Enumerating fiber bundles with fiber and base first Eilenberg-Maclane spaces

Question

How to enumerate fiber bundles (maybe, only as spaces, not bundles) with fiber $K(A, 1)$ and base $K(B, 1)$? It seems to be connected with enumerating short exact sequences of form $0 \to A \to ... \to B \to 0$, because such an exact sequence could induces something like "exact sequence" $K(A,1) \to K(..., 1) \to K(B, 1)$.

Answer 1

[Sorry, this is slightly too long for a comment]

Certainly if there exists a space $X$ and fibration $p\colon X\to K(B,1)$ with fiber $K(A,1)$ then there is an induced long exact sequence in homotopy given by $$\cdots\to 0 \to 0\to \pi_i(X)\to 0 \to \cdots \to 0 \to A \to \pi_1(X) \to B \to 0$$ which induces an isomorphism $0\stackrel{\cong}{\to} \pi_i(X)$ for all $i\geq 2$ and so $X$ must be a $K(\pi_1(X),1)$ which is an extension of $B$ by $A$. So yes, $0\to A\to \pi_1(X) \to B\to 0$ is a short exact sequence.

Whether all $K(G,1)$s for which $G$ is an extension of $B$ by $A$ can be realised (up to homotopy) as a bundle over $K(B,1)$ with fiber $K(A,1)$ is another question though.