Suppose we have a short exact sequence $1 \to K \to G \to Q \to 1$ and let $X_K$ and $X_Q$ be $K(K, 1)$ and $K(Q, 1)$, respectively. Is it possible to construct (in a "natural" way) a model of $K(G, 1)$ out of $X_K$ and $X_Q$?

For example, if $G$ is a direct product of $K$ and $Q$, then it acts on $\tilde{X}_K \times \tilde{X}_Q$ in a decent way. Is it possible to generalize it somehow?

I know that something can be done for cohomology (there is this Lyndon–Hochschild–Serre spectral sequence). But I need a topological space that is a model of $K(G, 1)$ (or rather, I am interested in the universal cover of that, and the action by deck transformations).