End AutIn.
Arguments Aut_in {gT} A%g B%g.
Section InjmAutIn.
Variables (gT rT : finGroupType) (D G H : {group gT}) (f : {morphism D >-> rT}).
Hypotheses (injf : 'injm f) (sGD : G \subset D) (sHG : H \subset G).
Let sHD := subset_trans sHG sGD.
Local Notation fGisom := (Aut_isom injf sGD).
Local Notation fHisom := (Aut_isom injf sHD).
Local Notation inH := (restr_perm H).
Local Notation infH := (restr_perm (f @* H)).

Lemma isom_restr_perm a : a \in Aut G -> fHisom (inH a) = infH (fGisom a).