Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $BH$?
In particular, we consider a split exact sequence $$ 1 \longrightarrow T^m \longrightarrow G_1 \longrightarrow F\longrightarrow1 $$ where $F$ is a finite group. It is well known that $BT^m=K(\mathbb Z^m,2)$ and $BF=K(F,1)$. How can we express $BG_1$？