Suppose a group $H$, not necessarily finite, acts on an Eilenberg-MacLane space $BN$. The homotopy quotient $BN/H$ (which agrees with the ordinary quotient if the action of $H$ is free) is an Eilenberg-MacLane space $BG$ for a group $G$ fitting into a short exact sequence
$$1 \to N \to G \to H \to 1$$
(determined by the action). In other words, it's an extension of $H$ by $N$. Every such extension arises in this way. Among them, semidirect products correspond to pointed actions of $H$ on $BN$, or equivalently actions of $H$ on $BN$ admitting a (homotopy) fixed point.
For example, take $BN$ to be the configuration space of $n$ ordered points in $\mathbb{R}^2$, so that $N = P_n$ is the pure braid group, and $H = S_n$ acts by permutations of the points. Then $BN/H = BG$ is the configuration space of $n$ unordered points in $\mathbb{R}^2$, so that $G = B_n$ is the usual braid group. The corresponding short exact sequence
$$1 \to P_n \to B_n \to S_n \to 1$$
does not split.