  In 1993, Muzychuk \cite{muzychuk} showed that the rational Schur rings over a
cyclic group $Z_n$ are in one-to-one correspondence with sublattices of the
divisor lattice of $n$, or equivalently, with sublattices of the lattice of
subgroups of $Z_n$. This can easily be extended to show that for any finite
group $G$, sublattices of the lattice of characteristic subgroups of $G$ give
rise to rational Schur rings over $G$ in a natural way. Our main result is that
any finite group may be represented as the (algebraic) automorphism group of
such a rational Schur ring over an abelian $p$-group, for any odd prime $p$. In
contrast, we show that over a cyclic group the automorphism group of any Schur
ring is abelian. We also prove a converse to the well-known result of Muzychuk
\cite{muzychuk2} that two Schur rings over a cyclic group are isomorphic if and
only if they coincide; namely, we show that over a group which is not cyclic,
there always exist distinct isomorphic Schur rings.
