Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Let $\mathrm{\Gamma }=\mathrm{C}\mathrm{a}\mathrm{y}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{\Gamma })$. The Cayley index of $\mathrm{\Gamma }$ is $\mathrm{\mid }A:G\mathrm{\mid }$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a noncyclic regular subgroup, then the Cayley index of $\mathrm{\Gamma }$ is superexponential in $p$. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if $p$ is an odd prime and $G$ is abelian but not cyclic, and has order a power of $p$ at least $p^3$, then there is a Cayley digraph $\mathrm{\Gamma }$ on $G$ whose Cayley index is just $p$, and whose automorphism group contains a nonabelian regular subgroup.