Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups).

Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$.

Write $G = G_1 \times G_2$, $M = M_1 \otimes M_2$ and $N = N_1 \otimes N_2$.

Is $Ext_G(M, N)$ related in some way to $Ext_{G_1}(M_1, N_1)$ and $Ext_{G_2}(M_2, N_2)$ (eventually with some extra conditions)?

In a more vague way, is there a way to "break down" $Ext_G(M, N)$?

Here I don't precise the degree of $Ext$, but $Ext^1$ will be of primary interest.