Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure on the locally compact Hausdorff group $G/H$.
My question is this: if $E$ is a union of cosets of $H$ in $G$ that is $\mu$-measurable (if necessary, $E$ can be assumed to be a Borel set); must then $q(E)$ be $\nu$-measurable?
I have seen work in descriptive set theory which shows that $q(E)$ is actually Borel whenever $E$ is a Borel $H$-invariant set, even when $H$ is not normal, but such work always assumes that $G$ is second countable. I am interested in locally compact Hausdorff groups that may not be second countable, and would be happy with $q(E)$ being just $\nu$-measurable.
If the answer is negative in general, does it hold at least if $G$ is abelian or compact?