Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting representation of $H$ on $H_{*}(G;\mathbb{Z})$, the integral homology of $G$. Is anything in general known about this $H$-representation? For instance, are there nice generators of the homology groups of $G$ on which I might try to describe the action of $H$? Also, I would be grateful for any and all references that might prove relevant.

Thanks!