Section ExternalAction.
Variables (pi : nat_pred) (aT gT : finGroupType).
Variables (A : {group aT}) (G : {group gT}) (to : groupAction A G).
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).

Lemma external_action_im_coprime : coprime #|G'| #|A'|.