Let $\mathrm{Aut}(\Gamma)$ be the automorphism group of a graph $\Gamma$. Also suppose that $\mathrm{Cay}(G)$$\mathrm{Cay}(G,S)$ is the Cayley graph of a group $G$ with respect to the generating set $S$. Consider the following chain:
$\Gamma \to \mathrm{Aut}(\Gamma) \to \mathrm{Cay}(\mathrm{Aut}(\Gamma)) \to \mathrm{Aut}(\mathrm{Cay}(\mathrm{Aut}(\Gamma))) \to \mathrm{Cay}(\mathrm{Aut}(\mathrm{Cay}(\mathrm{Aut}(\Gamma)))) \to \ldots$$\Gamma \to \mathrm{Aut}(\Gamma) \to \mathrm{Cay}(\mathrm{Aut}(\Gamma),S_1) \to \mathrm{Aut}(\mathrm{Cay}(\mathrm{Aut}(\Gamma),S_1)) \to \mathrm{Cay}(\mathrm{Aut}(\mathrm{Cay}(\mathrm{Aut}(\Gamma),S_1)),S_2) \to \ldots$ ,
where arrow $A\to B$ just means making $B$ from $A$. For example $\Gamma \to \mathrm{Aut}(\Gamma)$ means that the automorphism group $\mathrm{Aut}(\Gamma)$ is constructed based on $\Gamma$. Is there any graph $\Gamma$ for which the above chains stops, i.e. after $i$ steps
$\Gamma \cong \mathrm{Cay}(\mathrm{Aut}(\mathrm{Cay}(\cdots(\mathrm{Aut}(\Gamma))))$$\Gamma \cong \mathrm{Cay}(\mathrm{Aut}(\mathrm{Cay}(\cdots(\mathrm{Aut}(\Gamma))),S_i)$,
as two graphs (the notation $\cong$ means graph isomorphism)?
What about the case that starts with a group $G$, i.e. is there any group $G$ such that
$G \cong \mathrm{Aut}(\mathrm{Cay}(\mathrm{Aut}(\cdots(\mathrm{Cay}(G))))$$G \cong \mathrm{Aut}(\mathrm{Cay}(\mathrm{Aut}(\cdots(\mathrm{Cay}(G,S_1)),S_i))$,
as two groups (the notation $\cong$ means group isomorphism and $\mathrm{Aut}(G)$ is the automorphism group of the group $G$ here)?