On the number of subgroups of the group ${\mathbb{Z}}_{m_1}\times {\mathbb{Z}}_{m_2}$ with $m_1{m}_2\le x$ such that $m_1{m}_2$ is a $k$-th power

Let ${\mathbb{Z}}_m$ be the additive group of residue classes modulo $m$ and $s(m_1,m_2)$ denote the number of subgroups of the group ${\mathbb{Z}}_{m_1}\times {\mathbb{Z}}_{m_2}$, where $m_1$ and $m_2$ are arbitrary positive integers. We consider sums of type $\sum _{\begin{array}{c}{m}_1{m}_2\le x\\ m_1{m}_2\in N_k\end{array}}s(m_1,m_2)$, where $N_k$ is the set of $k$-th power of natural numbers. In particular, we deduce asymptotic formulas with $k=2$ and $k=3$.