Murder Mystery Question from an Entrance Exam

Here is a question I’ve been having some trouble with. I’d love any of your thoughts. It is Question 3 in this entrance exam.

Mr Cadbury was murdered last night.

Exactly one of six suspects is responsible, and each has made three
statements. Each has made at least one true statement.

Moreover, three of them like biscuits, and three do not, but you do not know which is which, only that people who like biscuits always
give an odd number of true statements, and people who don’t always
give an even number.

Here are the statements made:

Miss Burton:

• “Dr Lyons dislikes biscuits.”
• “Prof. Peek-Frean is the murderer.”
• “Col. Huntley-Palmer, Mrs McVitie and I are all innocent.”

Rev. Mr Fox:

• “Col. Huntley-Palmer killed him.”
• “Prof. Peek-Frean killed him.”
• “Miss Burton killed him.”

Col. Huntley-Palmer:

• “I don’t like biscuits.”
• “Neither Mrs McVitie nor Miss Burton likes biscuits.”
• “Prof. Peek-Frean and the Rev. Mr Fox both like biscuits.”

Dr Lyons:

• “I like biscuits.”
• “Mrs McVitie did not commit the murder.”
• “Miss Burton did not commit the murder.”

Mrs McVitie:

• “I did not commit the murder.”
• “Prof. Peek-Frean did not commit the murder.”
• “The Rev. Mr Fox dislikes biscuits.”

Prof. Peek-Frean:

• “I did not commit the murder.”
• “Dr Lyons and Miss Burton either both like biscuits or both dislike them.”
• “The murderer dislikes biscuits.”

I am able to show that the murderer must be one of Col. Huntley-Palmer (hereby referred to as H) and Prof. Peek-Frean (hereby referred to as P). I am able to show that H’s second statement H2 is false and that H3 is true. I also found out that P & F both like biscuits and that H & L don’t like biscuits. I am struggling with finding out the identity of the murderer.

Supposing that H is the murderer, I find that B likes biscuits (B has only one true statement) while M dislikes biscuits. Hence, P has two true statements, which contradicts our finding that P likes biscuits. This inconsistency forces us to conclude that P must be the murderer. But here too, there is a contradiction.

If P is the murderer then (I find that) B likes biscuits (B has three true statements) and so does M. (This is already a contradiction since there are only three people who like biscuits.) We further find that P has zero true statements, which is contradictory to the question’s premise that each person has at least one true statement.

Any ideas where I might have gone wrong? Thanks in advance!

Edit: I think the consensus is that the question has been incorrectly worded and as such, it has no solution. Thanks all.