Assume that finite groups $G_1$ and $G_2$ act smoothly on a manifold $M$ in such a way that the fixed point set, $M^{G_1\cap G_2}$, is an oriented closed manifold, $M^{G_1}$ and $M^{G_2}$ are its oriented closed submanifolds (they are orientable and we fix their orientations) and $M^{G_1}$ and $M^{G_2}$ are transverse in $M^{G_1\cap G_2}$ (all the assumptions are made so that we have a well-defined intersection number of $M^{G_1}$ and $M^{G_2}$ in $M^{G_1\cap G_2}$).
Assume that $M^{G_1}\cap M^{G_2}$ is a two-point set - $M^{G_1}\cap M^{G_2}=\{x_1,x_2\}$ and consider tangent spaces $T_{x_1}M^{G_1}$, $T_{x_1}M^{G_2}$ and $T_{x_1}M^{G_1\cap G_2}$. They are endowed with an $\mathbb{R}G_1$, $\mathbb{R}G_2$ and $\mathbb{R}(G_1\cap G_2)$-module structures respectively (analogously for the tangent spaces at $x_2$).
Is there a way to compute the intersection number of $M^{G_1}$ and $M^{G_2}$ in $M^{G_1\cap G_2}$ basing only on the $\mathbb{R}G$-module structures of the tangent spaces mentioned before (here $G\in\{G_1,G_2,G_1\cap G_2\}$)?