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.



$Ext$ functor over a product of groups
The answer is given by Kunneth formula. In your case for each pair of modules $N\_1$, $N\_2$ we can consider a functor from the category of representations of $G\_1\times G\_2$ to $\mathbb C$-vector spaces, sending $M$ to $\mathrm{Hom}\_{\mathbb C[G\_1\times G\_2]}(M, N\_1\otimes N\_2)$. $\mathrm{Ext}^\bullet\_{\mathbb C[G\_1\times G\_2]}(M\_1\otimes M\_2, N\_1\otimes N\_2)$ is its right derived functor computed on $M=M\_1\otimes M\_2$ and the computation can be done replacing $M\_1\otimes M\_2$ by a projective resolution. It is easy to see that if $P^\bullet$ and $Q^\bullet$ are projective resolutions of $M\_1$ and $M\_2$, then $P^\bullet\otimes Q^\bullet$ (this is bicomplex and I mean its totalisation) is a projective resolution of $M\_1\otimes M\_2$ and when you compute $\mathrm{Hom}\_{\mathbb C[G\_1\times G\_2]}(P^\bullet\otimes Q^\bullet, N\_1\otimes N\_2)$ you can check in particular that $\mathrm{Ext}^1\_{\mathbb C[G\_1\times G\_2]}(M\_1\otimes M\_2, N\_1\otimes N\_2)=\mathrm{Ext}^1\_{\mathbb C[G\_1]}(M\_1, N\_1)\otimes \mathrm{Hom}\_{\mathbb C[G\_2]}(M\_2, N\_2)\oplus \mathrm{Hom}\_{\mathbb C[G\_1]}(M\_1, N\_1)\otimes \mathrm{Ext}^1\_{\mathbb C[G\_2]}(M\_2, N\_2)$. 


In the case of finite groups categories of representations over $\mathbb C$ are semisimple and both sides are zero.


