Given a group action of $G$ on $M$, we have the notion of the stabilizer of a point $m$ :
$Stab_G(m)=\{g \in G | g(m)=m \}$
, and we could extend this to a definition of the stabilizer of a generic subset of $M$.
Reversing the roles, we could do something similar. Given a subgroup $H < G$, we could define a subset in $M$ s.t. :
$U_H=\{ m \in M | h(m)=m, \forall h \in H\}$,
i.e. the set of all elements which is stabilized by all elements of the subgroup $H$.
If we define $U_g$ as :
$U_g=\{ m \in M | g(m)=m\}$
than:
$U_H=\cap_{h\in H}U_{h}$
Do these definitions make sense ? Do they have a proper "official" name ?