right multiplication by elements of a discrete subgroup preserve left haar measure?

Question

If $\Gamma$ is a discrete subgroup of a locally compact topological group, G, it is not necessarily the case that right multiplication on $G$ by elements of $\Gamma$ will preserve a left Haar measure,correct? I'm confused because it sort of seems like $G/\Gamma$ should always carry a (left) G invariant Borel measure since $G$ covers $G/\Gamma$. But this is equivalent to $\Delta_G |_{\Gamma} =\Delta_{\Gamma}$ which says precisely that right multiplication by elements of $\Gamma$ preserves a left-Haar measure of $G$ (since $\Gamma$ is unimodular)