What do you call a group set up to undermine a political argument?

Another term, which is used particularly when the government in your question is the Russian or former Soviet government, is active measures. According to Wikipedia (the quote in the Wikipedia excerpt is from The Mitrokhin Archive):

Active measures range "from media manipulations to special actions involving various degrees of violence". They were used both abroad and domestically. They included disinformation, propaganda, counterfeiting official documents, assassinations, and political repression, such as penetration into churches, and persecution of political dissidents.

Example from Wikipedia:

Operation INFEKTION was a KGB disinformation campaign to spread information that the United States invented HIV/AIDS as part of a biological weapons research project at Fort Detrick, Maryland. (link to corresponding Wikipedia page)

In your case you could say those part of the government group to undermine the other group are members of active measures against that group.

Attribution:

1 "Active Measures." Wikipedia. April 04, 2018. Accessed April 04, 2018. https://en.wikipedia.org/wiki/Active_measures.

2 "Operation INFEKTION." Wikipedia. April 04, 2018. Accessed April 04, 2018. https://en.wikipedia.org/wiki/Operation_INFEKTION.

Just yesterday, lazily following the political developments in my country (Ukraine) where a lot of political "innovations", both home-grown and imported, are constantly being tested, I saw a similar phenomenon being called

fake activists.

Yes, that plain and simple.

All the suggestions made so far apply to situations that differ in important aspects from what you want to say.

A "false flag operation" is when a group of soldiers (or special forces operatives) of country A perpetrate a violent act making it look like they were in fact soldiers/operatives of country B. It has never (to my knowledge) been used for the situation that you are describing, which is a political trick restricted to civilian life. A classic example of a false flag operation is one that gave the Soviet Union a pretext to attack Finland in 1939, when "unknown operatives" shelled a Soviet border guard post, and this act was pinned on Finns (giving the Soviet Union the pretext to attack Finland), when in fact it was perpetrated by Soviet operatives.

"Th fifth Column" is a term amply present in our political discourse when discussing who is a "useful idiot" who, by his/her/their actions, benefits Russian interests, but again this is not used in discussions that do not involve an external enemy. The "fifth column" may not even exist—it might be an imaginary group of people that is accused of working for a different, purportedly hostile, country. E. g., this article from The New Yorker describes how President Putin of Russia sees people who are dissatisfied with his rule as people who try to sow discord and are thus working for the purportedly hostile West.

"Infiltrators" join a certain group—they don't set up a rival group that mimics someone.

Etc. etc.

People who pretend to be activists (of whatever movement) are just that — fake activists.

I'd probably call them entryists.

Slightly broader than you asked: (in US) when a powerful group (government, political, commercial, or religious) recruits or employs people to simulate popular support for the powerful group or its position(s) or proposal(s), usually but not inherently by opposing, criticizing, or interfering with the (other) people who criticize or oppose the powerful group, it is called Astroturf or Astroturfing (not always capitalized, sometimes hyphenated).

This is a metaphor: actual Astroturf is a brand of artificial grass widely used for sports facilities, and groups or movements truly made up of ordinary people are called 'grassroots' (sometimes hyphenated), so Astroturf-ers are artificial or fake versions of such groups and people.

I would call a person who works for the government and goes undercover in the group to undermine the group's authority or message an infiltrator, according to Cambridge Dictionary:

​a person who secretly becomes part of a group in order to get information or to influence the way the group thinks or behaves

The group itself could be referred to using the plural: infiltrators.

Attribution: "Infiltrator Meaning in the Cambridge English Dictionary." Cambridge Dictionary. Accessed March 31, 2018. https://dictionary.cambridge.org/dictionary/english/infiltrator.

Being Spanish, the expression Fifth column comes immediately to mind. It looks like it has been used in English, but I do not know how usual it is.

A name for the individuals in such a group, which could be pluralized for the group, agents provocateurs.

This is a False Flag operation, almost exactly by definition.

Maybe, we should call them saboteurs since their job is to sabotage the first group's cause? Here's the meaning of the verb to sabotage as defined by the Collins English Dictionary:

If someone sabotages a plan or a meeting, they deliberately prevent it from being successful.