Section AutIn.
Variable gT : finGroupType.
Definition Aut_in A (B : {set gT}) := 'N_A(B | 'P) / 'C_A(B | 'P).
Variables G H : {group gT}.
Hypothesis sHG: H \subset G.

Lemma Aut_in_isog : Aut_in (Aut G) H \isog restr_perm H @* Aut G.