Let $G$ be a group, $H$ a subgroup and $X$ a $G$-set. By taking orbits $X/H = X \times_H 1$ or fixed points $X^H = \mathrm{Hom}_H(1,X)$ we obtain a set on which $H$ acts trivially, and we've destroyed the $G$-action.

What we have left instead is an action of $N_H/H$, where $N_H$ denotes the normaliser of $H$, the subgroup $\{ g \in G : gH = Hg \}$.

What characterises this action abstractly?

Is there a universal property?

What other contexts does this construction exist in? Eg. rings $S \leq R$ and an $R$-module