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In this paper we use the Galois module structure for the classicalparameterizing spaces for elementary p-abelian extensions of a field K to givenecessary and sufficient conditions for the solvability of any embeddingproblem which is an extension of Z/p^nZ with elementary p-abelian kernel. Thisallows us to count the total number of solutions to a given embedding problemwhen the appropriate modules are finite, and leads to some nontrivial automaticrealization and realization multiplicity results for Galois groups.

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