Parameterizing solutions to any Galois embedding problem over Z/pnZ with elementary p-abelian kernel

In this paper we use the Galois module structure for the classical parameterizing spaces for elementary p-abelian extensions of a field K to give necessary and sufficient conditions for the solvability of any embedding problem which is an extension of Z/pnZ with elementary p-abelian kernel. This allows us to count the total number of solutions to a given embedding problem when the appropriate modules are finite, and leads to some nontrivial automatic realization and realization multiplicity results for Galois groups.