  We determine the Galois module structure of the parameterizing space of
elementary $p$-abelian extensions of a field $K$ when $\text{Gal}(K/F)$ is any
finite $p$-group, under the assumption that the maximal pro-$p$ quotient of the
absolute Galois group of $F$ is a free, finitely generated pro-$p$ group, and
that $F$ contains a primitive $p$th root of unity if $\text{char}(F) \neq p$.
