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].

(* Specifications: *)
Lemma restr_perm_on p : perm_on S (restr_perm p).
Lemma triv_restr_perm p : p \notin 'N(S | 'P) -> restr_perm p = 1.
Lemma restr_permE : {in 'N(S | 'P) & S, forall p, restr_perm p =1 p}.
Lemma restr_perm_Aut : restr_perm H @* Aut G \subset Aut H.
Lemma Aut_in_isog : Aut_in (Aut G) H \isog restr_perm H @* Aut G.
Lemma Aut_sub_fullP :
reflect (forall h : {morphism H >-> gT}, 'injm h -> h @* H = H ->
exists g : {morphism G >-> gT},
[/\ 'injm g, g @* G = G & {in H, g =1 h}])
(Aut_in (Aut G) H \isog Aut H).