If stealth aircraft are invisible to radar, then how does their own Air Force manage them or see them in flight?
Like most military aircraft, the F-22 has a military equivalent of a transponder, an IFF system (Wikipedia):

Identification, friend or foe (IFF) is a radar-based identification system designed for command and control. It uses a transponder that listens for an interrogation signal and then sends a response that identifies the broadcaster. It enables military and civilian air traffic control interrogation systems to identify aircraft, vehicles or forces as friendly and to determine their bearing and range from the interrogator.

Depending on mission profile the F-22 may also carry a radar cross section (RCS) enhancer to hide its stealth capabilities and make it more visible on radar.
Usually the main system for creating situational awareness during military operations and training is the Link 16 military tactical datalink. This system is carried by all modern NATO military aircraft, though F-22 will be getting the full feature Link 16 system including transmit functionality during 2021  . Link 16 also utilized by other branches of the military forces.