A reciprocity on finite abelian groups involving zero-sum sequences II

Let $G$ be a finite abelian group. For any positive integers $d$ and $m$, let $\varphi_G(d)$ be the number of elements in $G$ of order $d$ and $\mathsf M(G,m)$ be the set of all zero-sum sequences of length $m$. In this paper, for any finite abelian group $H$, we prove that $$|\mathsf M(G,|H|)|=|\mathsf M(H,|G|)|$$ if and only if $\varphi_G(d)=\varphi_H(d)$ for any $d|(|G|,|H|)$. We also consider an extension of this result to non-abelian groups in terms of invariant theory.