Asymptotic stability of Cayley graphs on abelian groups

For a finite group $G$, we say that a Cayley graph $Γ$ on $G$ is a most rigid representation (MRR) of $G$ if its full automorphism group has the smallest possible order among all Cayley graphs on $G$, and say that $Γ$ is stable if every automorphism of $Γ\times K_2$ comes from $\mathrm{Aut}(Γ)\times