So I'm curious about how the Cohen-Lenstra heuristics on ideal class groups might work over non-abelian extensions. Specifically:

Fix a finite group $G$ and suppose I have a family of Galois extensions of $\mathbf Q$ with Galois group $G$. Fix a prime $p$ not dividing #$G$. If $K$ is one of these extensions, then the $p$-torsion $C_K[p]$ in the ideal class group of $K$ is a semisimple ${\mathbf F}_p[G]$-module. How would one expect these to be distributed as $K$ varies?

An obvious guess is that for any finite ${\mathbf F}_p[G]$-module $M$, the (inverse of the) size of the ${\mathbf F}_p[G]$-automorphisms of $M$ would control the frequency with which $M$ occurs. However, I don't feel comfortable enough with the usual Cohen-Lenstra heuristics to have a feeling if there should be more to it than that.

Any suggestions and/or references? I've had no luck finding anything so far.