Free quotient group admits a section

Statement

Suppose $N$ is a normal subgroup of a group $G$ such that the quotient group $G/N$ is a free group.

Then, $N$ is a complemented normal subgroup of $G$ . In other words, there exists a retract $B$ of $G$ with normal complement $N$ , i.e., $B$ is a subgroup of $G$ such that $G$ is the internal semidirect product $N\rtimes B$ . Explicitly, $N\cap B$ is trivial and $NB=G$ .

A normal subgroup $N$ such that $G/N$ is a free group is termed a free-quotient subgroup.