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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) fits into a fiber sequence

$$1 \to BN \to BN/H \to BH \to 1$$

and the long exact sequence in homotopy shows that $BN/H$ has vanishing higher homotopy. Hence it 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.

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) fits into a fiber sequence

$$BN \to BN/H \to BH$$

and the long exact sequence in homotopy shows that $BN/H$ has vanishing higher homotopy. Hence it 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 (extensions for which the above sequence splits) 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.

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.

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) fits into a fiber sequence

$$1 \to BN \to BN/H \to BH \to 1$$

and the long exact sequence in homotopy shows that $BN/H$ has vanishing higher homotopy. Hence it 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.

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 $H = S_n$ acting on permutation in the obvious way. 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.

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.