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If $F : C \to D$ is any functor whatsoever, the objects $F(d)$ 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.