The usergroup field is an internal backend field. It does not display on frontend. It's typically not a good idea to expose to the public a users permission level. Usergroup field can only be edited from backend unless you use something like CB Auto Actions acting on a field value change to alter their usergroup.

Strange, the usergroup field is a part of the params field in CB > Field Management; ensure it hasn't been conditioned by CB Conditional or CB Privacy. Is your user a super user editing another super user or just regular registered users? Is the user you're editing synchronized? Try running User Synchronization within CB > Tools then retry editing them.