I am trying to solve the following exercise about classifying spaces (5.1.28) in the book "Algebraic K-Theory and its applications" by Rosenberg:
Let $$1 \longrightarrow N \longrightarrow G \longrightarrow \frac{G}{N} \longrightarrow 1$$ be a group extension. Show that there is a corresponding fibration of classifying spaces
$$BG \longrightarrow B\left(\frac{G}{N}\right)$$
with fibre $BN$.
I know that covering maps satisfy the homotopy lifting property for fibrations and I think I was able to show that $EG \longrightarrow BG \longrightarrow B\left(\frac{G}{N}\right) $ is a covering, which then allowed me to prove that the map is a fibration. However, I do not understand how I can now identify the fibre of this fibration to be (homotopy equivalent to) $BN$. Intuitively it makes sense that this is the case, but I have problems proving it. I have not much experience with fibrations and classifying spaces, so any help is greatly appreciated.