Recently, we have introduced a group invariant, which is related to the number of elements $x$ and $y$ of a finite group $G$ such that $x{wedge}y=1_{G{wedge}G}$ in the exterior square $G{wedge}G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m{wedge}k$ of $H{wedge}K$ such that $h^m{wedge}k=1_{H{wedge}K}$, where $m{geq}1$ and $H$ and $K$ are arbitrary subgroups of $G$.