Let $G$ be a finite group . Let $\Gamma$ be the graph $\Gamma=\operatorname{Cay}(G,H)$. Does $\operatorname{Aut}(\Gamma)$ contain an element of order $n=|G|$?
By $\operatorname{Cay}(G,H)$ I mean the Cayley graph of a group $G$ and connecting set $H$ and $\operatorname{Aut}(\Gamma)$ is the automorphism group of the graph $\Gamma$. We mean by "order of $a$" the least $n$ such that $a^n=1_{\operatorname{Aut}(\Gamma)}$.
My question is: when does $\operatorname{Aut}(\Gamma)$ contain a cyclic subgroup of order $n=|G|=|V(\Gamma)|$, when $G$ is nonabelian?