End QuotientAction.
Notation "to / H" := (quotient_action to H) : action_scope.
Section ModAction.
Variables (aT : finGroupType) (D : {group aT}) (rT : finType).
Variable to : action D rT.
Implicit Types (G : {group aT}) (S : {set rT}).
Section GenericMod.
Variable H : {group aT}.
Local Notation dom := 'N_D(H).
Local Notation range := 'Fix_to(D :&: H).
Let acts_dom : {acts dom, on range | to} := acts_act (acts_subnorm_fix to H).
Definition modact x (Ha : coset_of H) :=
if x \in range then to x (repr (D :&: Ha)) else x.

(* Specifications: *)
Lemma modactEcond x a :
a \in dom -> modact x (coset H a) = (if x \in range then to x a else x).
Lemma modactE x a :
a \in D -> a \in 'N(H) -> x \in range ->  modact x (coset H a) = to x a.