Let $G$ be a discrete subgroup of a connected finite dimensional Lie group $\Gamma$. Let $K$ be a maximal compact subgroup of $\Gamma$ and denote $X=\Gamma/K$. It is well-known that $G$ acts properly on $X$ via multiplication on the left.


Now suppose that a discrete group $\tilde{G}$ contains $G$ as a finite index subgroup (I'm not assuming that $\tilde{G}$ is a subgroup of $\Gamma$.)

When can the action of $G$ on $X$ be extended to a proper action of $\tilde{G}$ on $X$?
Can you give an example where this is not the case?


Thanks.