Example of non-preservation of isomorphism classes of $G$-sets under pullback along automorphisms of $G$, with $G$ simple?

Question

Definition: if $G$ is a group, $X$ is a left $G$-set, and $\rho$ is a group automorphism of $G$, let us say the data $(G,X,\rho)$ has property $P$ if $\rho^* X$ is not isomorphic to $X$ as a left $G$-set. (Here $\rho^*X$ is the left $G$-set with underlying set the same as for $X$, and with $g$ acting on $\rho^*X$ as $\rho(g)$ acts on $X$.)

I'm looking for examples of $(G,X,\rho)$ having property $P$; so far, the only example I've come up with is the following: if $H$ is some nontrivial group, if $Y$ is some set with free left $H$-action, if $G := H \times H$ as a direct product group with projection homomorphisms $H \xleftarrow{\pi_1} G \xrightarrow{\pi_2} H$ onto the 1st and 2nd components respectively, if $\rho$ is the automorphism of $G$ that exchanges the two $H$ factors (mapping each "identically" onto the other), if $X := \pi_1^* Y$, then $\rho^*X \simeq \pi_2^* Y$ and furthermore there is no $G$-equivariant map between $X$ and $\rho^*X$ (in either direction); in particular $X$ and $\rho^*X$ cannot be isomorphic as $G$-sets.

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Would anyone know how to find an example of $(G,X,\rho)$ having property $P$ where $G$ is a simple group? (Or, how to show such an example doesn't exist?)

Answer 1

If $\rho$ is an automorphism and $H$ is a subgroup of $G$, then $\rho^* (G/H) \cong G / \rho^{-1}(H)$. This is isomorphic to $G/H$ iff $H$ and $\rho^{-1}(H)$ are conjugate. This yields many examples where they are not isomorphic. One case is when $G$ is abelian and $\rho(H) \neq H$.