  We review the basic theory of super $G$-spaces. We prove a theorem relating
the action of a super Harish-Chandra pair $(G_0, \mathfrak{g})$ on a
supermanifold to the action of the corresponding super Lie group $G$. The
theorem was stated in [DM99] without proof. The proof given here does not use
Frobenius theorem but relies on Koszul realization of the structure sheaf of a
super Lie group (see [Kosz83]). We prove the representability of the stability
subgroup functor.
