Let's say there is a guy who wants to be part of some group of people. And although is let to know that he is unwanted, he is annoyingly trying to join the people. What is the most common verb for it?
I am unaware of any formal word for such behavior, but informally the person will be be to said to "butt in" or "horn in".
(Typically, "butt in" means the person is participating without being wanted, and thereby obstructing the function of the group — a person who butts into a conversation by interrupting. "Horning in" implies that the person is not just obstructing the group but actually misappropriating some of the benefits of the group — if you horn in on a picnic, you are eating some of the food.)