Variable sH : socleType rH.
Definition Clifford_act (W : sH) x :=
let Gx := subgP (subg G x) in
PackSocle (component_socle sH (Clifford_simple (socle_simple W) Gx)).
Let valWact W x : (Clifford_act W x :=: W *m rG (sgval (subg G x)))%MS.
Fact Clifford_is_action : is_action G Clifford_act.
Definition Clifford_action := Action Clifford_is_action.
Local Notation "'Cl" := Clifford_action (at level 8) : action_scope.

Lemma Clifford_astab : H <*> 'C_G(H) \subset 'C([set: sH] | 'Cl).