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Let $\Gamma=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let$A=\mathrm{Aut}(\Gamma)$. The Cayley index of $\Gamma$ is $|A:G|$. It haspreviously 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 $\Gamma$ issuperexponential in $p$.  We present evidence suggesting that cyclic groups are exceptional in thisrespect. Specifically, we establish the contrasting result that, if $p$ is anodd prime and $G$ is abelian but not cyclic, and has order a power of $p$ atleast $p^3$, then there is a Cayley digraph $\Gamma$ on $G$ whose Cayley indexis just $p$, and whose automorphism group contains a nonabelian regularsubgroup.

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