Depends on the avionic fit of the aircraft as to how you enter into the transponder - this is my understanding, but not an expert on how data gets into the transponder:
a) If smaller GA type transponder, can be entered via the controls on the front panel. This should be the aircraft registration (e.g. G-ABCD) unless an ICAO callsign has been allocated, in which case that should be entered.
b) larger commercial aircraft normally have a control panel for either the FMS or the transponder, allowing the pilot to enter the ICAO callsign at the beginning of the flight prior to pushback. It may be that it can be automatically transferred to the transpodner via the FMS.
However it is done, the Aircraft ID (either registration or ICAO callsign) populates Comm-B register BDS 2,0 using IA-6 encoding as per Annex 10 Volume 4. It is a European requirement for all Mode S transponders to be able to supply this register containig the Aircraft ID.
The ground Mode S interrogator will request the BDS 2,0 register automatically during first detection into radar cover. Once received, the ground interrogator will not ask for it again unless the track is lost then re-acquired, or the transponder indicates that the Aircraft ID has changed.
Hope this helps,
RS