Catching a thief in a closed circle of people I am trying to investigate the issue with money getting lost form peoples wallets in a shared flat.

There are 4 suspects for being a thief of the money. Some of the suspects have been affected as well - the money was stolen from them. This complicates the issue, cause I believe, from the psychological point of view, some people may report that their money were stolen from them as well, just to get out of the picture. The question is - is there a psychological approach as to how to find the person behind this? Are there any methods how can one lead the conversation with the whole group to slowly navigate a thief to confess directly or indirectly through guiding questions or perhaps some bluffing? Are there signs of behaviour to look for when leading the discussion?
One way to get at this is to use the "guilty knowledge" test. https://en.wikipedia.org/wiki/Lie_detection#The_control_question_test_and_the_guilty_knowledge_test

In this test, you would need to first find out exactly how much money was reported stolen, and from whom. Then you would ask each suspect a combination of "true" questions and "false" questions. A true question contains only true details of the incident, while a false question contains some false details of the incident.

Let's say Tom, Dick, and Harry claim to have had 10, 12, and 20 dollars stolen from them respectively. If Tom did it, he is lying about his amount, so during his turn, you would ask him ``Did you steal 12 from Dick and 20 from Harry, for a total of 32 dollars?'' You would also ask him some "false" versions of the question, e. g., "Did you steal 11 dollars from Dick and 2 dollars from Harry, for a total of 13 dollars?" You would then see if he reacted differently to the "true" version of the question compared to the "false" version of the question. If he does, then it could be due to him knowing how much was stolen, and that would be evidence he was the one who stole it. You could enhance the effect by monitoring heart rate (perhaps with a phone app) throughout the questioning.

For this to work, it must be the case that you and the thief both know the exact amount stolen, but that none of the victims know this information because they haven't talked to each other. The thief would know because s/he stole it, and you would know from surveying everyone and adding it all up.

Additionally, you can find out when each person's money went missing, where the wallet was located during each theft, and what denominations the bills were. Include these details in your "true" questions, and change them in your "false" questions.

Even then, this is an unreliable test to implement on your own, so there's no guarantee it would work. Most likely you're out of luck.