What is the universal property of the Weyl group?

Question

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.

Answer 1

The group $WH$ is the automorphism group of $G/H$ as a $G$-set. Thus, $X^H$ is the set of morphisms from $G/H$ into $X$ in the category of $G$-sets, and this has an obvious action by precomposition with automorphisms.

Answer 2

If $F : C \to D$ is any functor whatsoever, the objects $F(c), c \in C$ always have a natural action of the automorphism group $\text{Aut}(F)$ of $F$ as a functor, and $\text{Aut}(F)$ is universal with respect to this property. If $F : C \to \text{Set}$ is a representable functor, this automorphism group coincides with the auotmorphism group of the representing object, by the Yoneda lemma. The functor of taking $H$-fixed points is representable by $G/H$, whose automorphism group is the Weyl group.

In general, it's an interesting question to ask what extra structure some objects $d \in D$ acquire by virtue of having been spit out by a functor $F : C \to D$. A very general answer to this question is that under mild hypotheses $F$ has what's called a codensity monad, and the objects $F(c), c \in C$ naturally acquire the structure of an algebra over this monad. If $F$ is the right adjoint of an adjunction this is the usual monad arising from that adjunction.

If $F : C \to \text{Set}$ is a representable functor represented by an object $c$, it has a left adjoint given by sending a set $X$ to the coproduct $\coprod_X c$ of $X$ copies of $c$ (assuming that these coproducts exist), and the codensity monad sends a set $X$ to the set $\text{Hom}(c, \coprod_X c)$. If $c$ is "connected" in the sense that a morphism $c \to \coprod_X c$ factors through one of the inclusions of a copy of $c$ (which is the case here, where $c = G/H$), then this is the monad whose algebras are $\text{End}(c)$-sets; otherwise it's more complicated.