Let $G$ be a topological group, let $A,B$ be closed subgroups of $G$ such that $A$ is an open subgroup in $B$. Then we have an open continuous map $f:G/A\to G/B$, with typical fiber $B/A$. I read the claim that $f$ is a covering map (which sounded very convincing at first) but I was unable to prove this.

This is easy if $B$ is a discrete subgroup, and I would be happy with a proof that works if $B$ is locally compact.

So what can be said about $f$ in general?

(Everything is Hausdorff.)