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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 $H$?

Your answer will be much appreciated.

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Assume you have the exact sequence $$1\rightarrow H\rightarrow G\rightarrow G/H\rightarrow 1$$Then it induces a fibration $$BH\rightarrow BG\rightarrow B(G/H)$$ as we imagine some large enough total space $EG$ whose quotient by $G$ is $BG$, by $H$ is $BH$, etc.

Now assume we know $B(G/H)$ and $B(H)$ but do not know $BG$, then we need certain invariants to distinguish $BG$ from various possible fibrations that are not isomorphic. Without further information of the structure of $BH$ and $B(G/H)$, this problem is as hopeless as classifying any fibrations over any base space. This is because we have $$B\Omega X=X$$for $X$ a connected topological space with a fixed base point. So in principle we do not know.

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