Let $G$ be a topological group and $H$ be a normal subgroup of $G$ (I think $H$ is required to be admissible in the sense that the quotient map $G\to G/H$ is a principal $H$-bundle, am I right?). Then there exists a homotopy fibre sequence $BH\to BG\to B(G/H)$, where $BG$ denotes the classifying space of $G$.

My questions is: suppose that we already know the groups $G$ and $H$ and suppose that we know the classifying space of $G/H$ and the classifying space of $H$, to what extent can we decide the classifying space of $G$ from these information? How to find the classifying space of $G$ if we know the classifying space of $G/H$ and the classifying space of $H$?

Any reference will be greatly appreciated.