Section FullExtension.
Local Notation inA := (sdpair2 to).
Local Notation inG := (sdpair1 to).
Local Notation A' := (inA @* gval A).
Local Notation G' := (inG @* gval G).
Let injG : 'injm inG := injm_sdpair1 _.
Let injA : 'injm inA := injm_sdpair2 _.
Hypotheses (coGA : coprime #|G| #|A|) (solG : solvable G).
End FullExtension.

(* We only prove a weaker form of the coprime group action centraliser  *)
(* lemma, because it is more convenient in practice to make G the range *)
(* of the action, whence G both contains H and is stable under A.       *)
(*   However we do restrict the coprime/solvable assumptions to H, and  *)
(* we do not require that G normalize H.                                *)
Lemma ext_coprime_quotient_cent (H : {group gT}) :
H \subset G -> [acts A, on H | to] -> coprime #|H| #|A| -> solvable H ->
'C_(|to)(A) / H = 'C_(|to / H)(A).