  We show that for certain integers $n$, the problem of whether or not a Cayley
digraph $\Gamma$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of
some other abelian group $G$ of order $n$ reduces to the question of whether or
not a natural subgroup of the full automorphism group contains more than one
regular abelian group up to isomorphism (as opposed to the full automorphism
group). A necessary and sufficient condition is then given for such circulants
to be isomorphic to Cayley digraphs of more than one abelian group, and an
easy-to-check necessary condition is provided.
