What options does a lawyer have if a client admits he is guilty but doesn't want to admit it to the court?

It is not the duty of the defense counsel to determine the guilt or innocent of his client. For as defense counsel, it is legally and ethically defend his client regardless of his guilt of not which is beyond the job of the lawyer to determine. A defense lawyer is bound to use any available legal remedy and use those remedy vigorously to obtain a reasonable doubt in favor of his client. Beside, a person criminally charge are always presume innocent hence it is the government prosecutor job to establish the accused guilt beyond reasonable doubt and that is the job of the defense counsel. Moreover, in some cases, the defense counsel can obtain a lesser conviction from the original charged against his client.

In the guilt or innocence phase of the case (which is really not the one where a lawyer is likely to be the most effective in most cases like this one), the primary strategy is to force the prosecution to prove the case beyond a reasonable doubt and to point out at trial every way that the evidence fails to do so.

For example, Colorado Rule of Professional Conduct 3.1 (which has an almost identical counterpart in every U.S. state) says:

Rule 3.1. Meritorious Claims and Contentions

A lawyer shall not bring or defend a proceeding, or assert or controvert an issue therein, unless there is a basis in law and fact for doing so that is not frivolous, which includes a good faith argument for an extension, modification or reversal of existing law. A lawyer for the defendant in a criminal proceeding, or the respondent in a proceeding that could result in incarceration, may nevertheless so defend the proceeding as to require that every element of the case be established.

OFFICIAL COMMENTARY

1 The advocate has a duty to use legal procedure for the fullest benefit of the client’s cause, but also a duty not to abuse legal procedure. The law, both procedural and substantive, establishes the limits within which an advocate may proceed. However, the law is not always clear and never is static. Accordingly, in determining the proper scope of advocacy, account must be taken of the law’s ambiguities and potential for change.

[2] The filing of an action or defense or similar action taken for a client is not frivolous merely because the facts have not first been fully substantiated or because the lawyer expects to develop vital evidence only by discovery. What is required of lawyers, however, is that they inform themselves about the facts of their clients’ cases and the applicable law and determine that they can make good faith arguments in support of their clients’ positions. Such action is not frivolous even though the lawyer believes that the client’s position ultimately will not prevail. The action is frivolous, however, if the lawyer is unable either to make a good faith argument on the merits of the action taken or to support the action taken by a good faith argument for an extension, modification or reversal of existing law.

[3] The lawyer’s obligations under this Rule are subordinate to federal or state constitutional law that entitles a defendant in a criminal matter to the assistance of counsel in presenting a claim or contention that otherwise would be prohibited by this Rule. See A.L.L. v. People ex rel. C.Z., 226 P.3d 1054, 1060 (Colo. 2010) (addressing obligations of court-approved counsel for a respondent parent in a termination of parental rights appeal).

Any evidence which tends to be exculpatory could also be introduced. For example, perhaps a defense lawyer could find a forensic science article showing that the technique used to link the bullet to the gun by police investigators is unreliable. Or, perhaps there is alibi evidence putting your client on the other side of town at the time that the coroner erroneously believes the time of death to have occurred.

The lawyer would almost always advise the client not to talk to anyone about the case in any respect, so as not to create evidence which could be used against him. Usually, in a case like this one, the safer bet would be to not have the defendant testify (which is by far the more common choice in most criminal cases). If the client testifies, a judge or jury could say that they found him guilty because his demeanor demonstrated guilt and it seemed as if he was lying, but if the client doesn't testify, that kind of finding of fact can't support a conviction.

Any evidence which is not admissible would be objected to (e.g. evidence collected in violation of the 4th Amendment).

In the typical exclusionary rule case (usually involving drugs or other contraband), the cops, the lawyers for both sides and the judges all know that your client is guilty, and a defense lawyer is still entitled to try to exclude the evidence and exonerate the client at trial when it was illegally obtained.

Criminal defense lawyers are not barred from arguing false inferences when representing their clients in criminal cases, they are only barred from offering perjured testimony.

Must the lawyer admit it to the judge or is he not allowed to do so, and still must help the client?

The lawyer is not allowed to do so and would probably be disbarred if he did without informed client permission.

The lawyer must still help the client.

Just because the defendant is factually guilty doesn't mean that the prosecution has sufficient evidence to prove every element of the offense beyond a reasonable doubt.

If the prosecution can easily prove the case, however, the lawyer's best available course of action is best described in this link, although a client will not always agree to this approach.

The defense cases that are ridiculed on TV or in news reports (e.g. the "Twinkie defense") usually involve guilty defendants who aren't willing to plea guilty whose lawyers come up with the most plausible possible defense under the circumstances, even if it sucks and isn't very likely to be believed. That comes with the territory.

This is the whole reason that Attorney Client Privilege exists. So that that the client can tell his/her attorney everything and not create a burden of admitting to a crime they commit. The Defense's job isn't to find the truth, it's to get the Defendant off of as many charges as he can. The Defendant may be guilty as sin and did everything he was charged with, but proving that he didn't do it isn't the attorney's tactic... he needs to show that the prosecution cannot say 100% certainty that the defendant did it alone.

For a good view for how this works see the movie "My Cousin Vinny". The titular character is a freshly minted lawyer who's background and rookie mistakes leaves the judge less than impressed (He's played by Joe Pesci in full Joe Pesci mode). In a moment of doubt, the defendant, Vinny's cousin, expresses his doubt in Vinny's ability to convince the jury they didn't do it. Vinny even says that everyone in the court room thinks that the accused is guilty. But Vinny demonstrates through a metaphor, then later shows, is that the case the prosecutor is bringing fourth relies on unreliable witnesses who's seemingly solid story's confirming the defendant was there at the time of the crime were... not the best witnesses (and what's brilliant is that prior to trial, Vinny questions all three witnesses prior to trial and demonstrates why there are no such things as stupid questions.), (additionally we the audience know from the outset that the defendant is innocent, but the point is that the evidence still points to their guilt.).

In real life, while they are legally innocent, O.J. Simpson and Casey Anthony are two individuals who are widely agreed by the public to have gotten away with the crimes they are accused of. In O.J.'s case, the Defense relied on some missteps in the investigation and the public ignorance of DNA profiling to show so doubt that O.J. did it (The People vs. O.J., which dramatized the trial, plays the later for all it's worth, initially framing DNA in a way a contemporary audience would recognize as standard police show dialog... only for another character to ask what DNA even meant... and a few of the other people in the room react like this is a reasonable question to bring up.).

In Casey Anthony case, Anthony was by the time of trial such a notorious liar, that the Defense merely had to point out that there was too many stories and the evidence doesn't prove who did what when to any degree of certainty. Basically so many narratives had been thrown out in such a high profile case, the evidence couldn't disprove all the lies or even establish a workable timeline of events.

The position is similar in England and Wales (note that Scotland and Northern Ireland are different jurisdictions with different rules). Lawyers in England and Wales have, in essence, two duties:

1. A duty to the court
2. A duty to their client

A lawyer's duty to their client - which includes their duty of confidentiality to that client - overrides everything except their duty to the court. See, for example, the current Bar Code of Conduct for barristers (contained in the BSB handbook, link here):

CD1 You must observe your duty to the court in the administration of justice [CD1] [...] CD1 overrides any other core duty, if and to the extent the two are inconsistent.

Similarly, from the SRA Code of Conduct for solicitors (contained within the SRA handbook, link here):

You must achieve these outcomes:

O(5.1) you do not attempt to deceive or knowingly or recklessly mislead the court [...]

O(5.5) where relevant, clients are informed of the circumstances in which your duties to the court outweigh your obligations to your client

This means that if your client tells you they are guilty, you cannot tell the court, as this would breach your duty to your client. For example, the Bar Code of Conduct, at rule C3.5, states:

Your duty to the court does not require you to act in breach of your duty to keep the affairs of each client confidential.

However, neither can you allow your client to say that they did not commit the offence, or to suggest that someone else did commit the offence, in any of the proceedings, as this would breach your duty to the court. Rule rC6 of the Bar Code of Conduct states that you cannot call witnesses to give evidence, or permit witness statements which give evidence, which you know to be untrue or misleading.

The solution to this is often to put the prosecution to proof. Under English law, the defence does not need to make a case: the accused is presumed to be innocent, and it is for the prosecution to prove otherwise. Thus if a lawyer knows their client is guilty, one solution is to not produce any defence evidence (as this could lead to perjury or misleading the court), but to leave the prosecution to make out their case.

In the Bar Code of Conduct, Guidance gC9 and gC10 clarifies the extent to which you can still continue with your case, and the actions you are and are not permitted to take, in the scenario in which your client tells you that they have committed the crime they are charged with.

If a client insists on continuing to run their case, and as part of that insists that - for example - they will give evidence to the effect that they did not commit the crime, the lawyer must then stop acting for the client.

Since this question does not assume jurisdiction, I will answer for the USA, but do know in many other common law jurisdictions (particularly the UK) the response suggested in my answer may be wrong and may even lead to contempt, disbarment or charges against a lawyer who performs them.

In the USA, a lawyer has an ethical responsibility not to intentionally elicit perjured testimony. Typically, if a lawyer knows a witness perjured themselves, they have an obligation to inform the court or impeach the witness. The big exception to this is if the lawyer knows about the perjury from a privileged communication with their client. Then the obligation of the lawyer is to not elicit any more perjured testimony and not to reveal the privileged information to the court — instead, they may state to the court something along the lines of "ethical rules prevent me from continuing this line of questioning" (of course, not necessarily that exactly, but something similar to that). However, the lawyer also has a constitutional obligation to zealously defend their client. That means if a lawyer's client wishes to perjure themselves on the stand, the lawyer cannot reveal that to the court, but they cannot aid that perjury either. In such a case, a lawyer will typically move to be allowed to step off the case or, if a judge denies that request, they will typically ask their client to testify in a narrative form once they arrive at the portion of perjured testimony.

However, the lawyer can still pursue a line of defense based on testimony they don't know is perjured even if the conclusions from that testimony are false. For example, if a client wants to tell the court that he has an alibi — for example, a client is charged with murder and he claims that he was on the other side of town at a restaurant when the murder was committed, yet the client actually told the lawyer that was a lie — the lawyer in that case would still have an obligation to present the testimony of a witness who believes they saw their client at that restaurant (a case of mistaken identity on the witness perhaps, but not a product of perjury).

So in conclusion, there are many options left for the lawyer, but most choices leave the lawyer in at least something of an ethical quandary.