End ActpermOrbits.
Section RestrictPerm.
Variables (T : finType) (S : {set T}).
Definition restr_perm := actperm (<[subxx 'N(S | 'P)]>).
Canonical restr_perm_morphism := [morphism of restr_perm].

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