ply: subset_trans trHK; rewrite subsetI subsetIr -{2}(mulg1 H) -mulGS.
  rewrite setIC group_modl ?joing_subr //= cent_joinEr // -eHK.
  by rewrite -group_modr ?joing_subl //= setIC -(normC (sub1G _)) mulSg.
apply: subset_trans trGH; rewrite subsetI subser_act // /restrm /= !invmE ?actf ?gact_stable.
Qed.

(* Specifications: *)
Qed.