Are there conditions that guarantee that a split short exact sequence of groups $$ 1 \rightarrow K \rightarrow G \rightarrow Q \rightarrow 1 $$ gives rise to a fiber bundle $$ F \rightarrow E \rightarrow B $$ on the level of $K(\pi,1)$'s?
Any references relating fibrations to group extensions would be appreciated.
Let $1 \to H \to G \to K \to 1$ be an exact sequence of topological groups. If $G \to G/H$ is a principle $H$-bundle, then theorem 11.4 here guarantees a fibration of classifying spaces $BH \to BG \to BK$. This assumption is satisfied if, for instance, $G$ is discrete, or if all the groups involved in the sequence are Lie groups.