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 ext_coprime_Hall_trans (H1 H2 : {group gT}) :
pi.-Hall(G) H1 -> [acts A, on H1 | to] ->
pi.-Hall(G) H2 -> [acts A, on H2 | to] ->
exists2 x, x \in 'C_(G | to)(A) & H1 :=: H2 :^ x.