If anyne could give me some help with this, it will be deeply appreciated:
Let $X$ be a set equipped with the action of some group $G$. Denote by $Aut_G(X)$ the set of $G-equivariant$ bijections $f: X \to X$ and take for granted that this is a group under composition.
If we let $H$ be a subgroup of $G$ and let $X = G/H$ equipped with the usual G-action (i.e. left multiplication), is it possible to find an isomorphism between $Aut_G(X)$ and $N_G(H)/H$, with $N_G(H)$ being the normalizer of $H$?
Thank you very much in advance for any help.