Let $G$ be a locally compact group and let $H$ be a cocompact (or more generally, a cofinite) subgroup of $G$.

Is there any criterion to determine whether $H$ contains a cocompact normal subgroup of $G$? More specifically, when does $G$ possess a normal cocompact subgroup? (regardless of $H$)

Is there a non-obvious example (e.g. $G$ non-abelian and non-compact) of this phenomenon?

P.S. Feel free to restrict your answer to Lie groups if it helps.