If $G$ is a group and $H\le G$ a subgroup, let $NH$ denote the normalizer of $H$ in $G$, and let $WH = NH/H$; following May's *Concise Course*, §3.4 this I call the *Weyl group*. I have also seen the definition $WH=NH/C(H)$, where $C(H)$ is the centralizer of $H$ in $G$, from https://groupprops.subwiki.org/wiki/Weyl_group.

What is the universal property of $WH$? It would also be good to have a notion of $Wf$ for any morphism $H \mathop{\longrightarrow}\limits^f G$, rather than just inclusions.

I'm asking this in pursuit of an abstract-nonsensical explanation of:

If $H\le G$ is a subgroup, and $X$ is a $G$-set, then $X^H$ is naturally a $WH$-set.

For example, since much of the formalism of $G$-sets extends to $\mathbf{Set}^C$ for any small category $C$, I might eventually try to generalize this to any functor between small categories.