13.4 (1995)

Let $G$ be a group with a normal (pro-) 2-subgroup $N$ such that $G/N$ is isomorphic to $GL_n(2)$, inducing its natural module on $N/\Phi(N)$, the Frattini factor-group of $N$. For $n = 3$ determine $G$ such that $N$ is as large as possible. (For $n > 3$ it can be proved that already the Frattini subgroup $\Phi(N)$ of $N$ is trivial; for $n = 2$ there exists $G$ such that $N$ is the free pro-2-group generated by two elements).