  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 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 $\Gamma$ 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 $\Gamma$ on $G$ whose Cayley index
is just $p$, and whose automorphism group contains a nonabelian regular
subgroup.
