quotient of fiber bundles

Question

Let $G$ a topological group and consider $P$ a $G$-fiber bundle over $B$. Let $H$ any subgroup of $G$ and let $Q$ an $H$-fiber bundle over $B$. Let $P/Q$ be quotient over $B$, such that the fiber has $G/H$ structure (no necessary a group structure). Is there any classifying space for this kind of quotient?