I am the site admin and there is a parmission group of the users with "contribute" level. At the group settings I see an option "Who can edit the membership of the group?". I set it to "Group Members". But even if a member of the group opens the site setting there is no options to edit any permission on the site. How can they add new members? Do they have to have "Full control" to be able to do that? Or I am just overlooking something and it is possible to let the group members manage their own group's memberships by themself even if they have only contribure level of control? If yes, how can they add or remove members?
Add a copy of Contribute permission level with adding
Enumerate Permissions option. Change Contribute permission of your group to the new one.
Look for details: Site permissions