Let G be a transitive group of odd prime-power degree whose Sylow p-subgroup P is abelian with t elementary divisors. We show that if p > 2^{t − 1}, then G has a normal subgroup that is a direct product of t permutation groups of smaller degree that are either cyclic or doubly-transitive simple groups. As a consequence, we determine the full automorphism group of a Cayley digraph of an abelian group with two elementary divisors such that the Sylow p-subgroup of the full automorphism group is abelian.