What do they usually say or ask when the fighter pilot wants to take off, when it's in the air, and what is the reply from the fighter pilot?
closed as too broad by fooot♦, Ralph J, Sean, bogl, J. Hougaard Feb 11 at 11:53
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This ICAO document lays out the guidelines pretty well in short, in most places military aircraft (of any kind) are treated no different than civilian aircraft and are expected to comply with local regulations which include radio procedures. ATC will clear an F-22 for take off no differently than a Cessna 150 (with perhaps different expected climb rates ;) )
Of course most jurisdictions that belong to countries that have some kind of an Air Force will generally coordinate with them for special types of operations and the organization that controls the airspace expects that anyone wanting to conduct special operations will coordinate with them.
This typically happens in controlled areas like an MOA or for safer operations near a base. I fly quite a bit near a large base and often while transitioning through their airspace its not uncommon to get warnings of high speed or low altitude activity. Either way the aircraft is expected to make radio calls and adhere to any regulations or pre cleared operations.