In 1993, Muzychuk showed that the rational S-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 idea is easily extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational S-rings over G in a natural way. Our main result is that any finite group may be represented as the automorphism group of such a rational S-ring over an abelian p-group. In order to show this, we first give a complete description of the automorphism classes and characteristic subgroups of finite abelian groups. We show that for a large class of abelian groups, including all those of odd order, the lattice of characteristic subgroups is distributive. We also prove a converse to the well-known result of Muzychuk that two S-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 S-rings. Finally, we show that the automorphism group of any S-ring over a cyclic group is abelian.