If $G$ is a group and $H$ a subgroup which is not normal. What is the homologies of the action of $G$ on the left coset space $G/H$? The action is multiplication from left.
More precisely, $G/H$ is a set that $G$ acts on it by permutation. Consider $M=\mathbb{Z}^I$ such that $I$ is the set of left cosets $G/H$. $G$ is acting on $M$ by permutation on the factors (like a potentially infinite dimensional permutation matrix.) Now you can look at $M$ as a $\mathbb{Z}[G]$ module. What is its homology groups.
You're asking what $H_*(G, \mathbb{Z}[G/H])$ is. Note that $\mathbb{Z}[G/H] = \mathbb{Z}[G]\otimes_H \mathbb{Z}$ where $\mathbb{Z}$ on the left is given the trivial $H$-action, so that $\mathbb{Z}[G/H] = \mathrm{Ind}_H^G\mathbb{Z}$; therefore by Shapiro's lemma,
$$H_*(G,\mathbb{Z}[G/H]) = H_*(H,\mathbb{Z})$$
Once you have this, you'll see that we can't be more precise than that because it would imply knowing the homology of $H$, which is not known in general