Consider a locally compact group $G$ and a cocompact subgroup $H$, is it known that the induction of an irreducible representation $\pi$ of $H$ to $G$ decomposes discretely into a direct sum of irreducible with finite multiplicity?

I know that this is true, for $\pi$ trivial and $H$ unimodular (=> G unimodular).

Consider a locally compact group $G$ and a cocompact subgroup $H$, is it known that the induction of an irreducible representation $\pi$ of $H$ to $G$ decomposes discretely into a direct sum of irreducible with finite multiplicity?

I know that this is true, for $\pi$ trivial and $H$ unimodular (=> G unimodular). Is unimodularity here necessary?

Please provide either a counterexample or a reference, that this does or does not hold in general.