The Eilenberg-MacLane spaces $K(G,q)$ are readily generalized to study cohomology with local coefficients.The generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ are spaces with only two nnvanishing homotopy groups, one of them the fundamental group is $\pi$, the other is $G$ in dimension $q$. And the fundamental group $\pi$ acts on $G$ by representation $\rho:\pi\rightarrow Aut(G)$, where $Aut(G)$ is the group of automorphisms of $G$.

It has been given the construction of Eilenberg-MacLane space K(G,n) in the category of CW complex in All Hathcer's book 'ALGEBRAIC TOPOLOGY'. In Samuel Gitler's article 'Cohomology Operations with Local Coefficients' . The generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ are constructed in the category of Kan Complex, but I have no idea about Kan complex and I have not learned them before. Is it possible to reference me some books or articles which talk about $K_{\pi}(G,q)$ in the category of CW complex?

And I have some idea about construction in the follow. But I am not sure if I am right.

Let $K(\pi,1)$ be the base space and homomorphism $\rho:\pi\rightarrow Aut(G)$ be the representation. The homomorphism $\rho$ can induce representation $\rho_{*}:\pi\rightarrow Homeo(K(G,q);K(G,q))$, where $Homeo(K(G,q);K(G,q))$ is the group of homeomorphisms of $K(G,q)$ itself. Then we can construct a fiber bundle $K(G,q)\rightarrow E\rightarrow K(\pi,1)$ by the representation $\rho_{*}$.

I want know if the total space $E$ is the generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ we need?