5.35 (1976)

Let $V$ be a vector space of dimension $n$ over a field. A subgroup $G$ of $GL_n(V)$ is said to be rich in transvections if $n \geqslant 2$ and for every hyperplane $H \subseteq V$ and every line $L \subseteq H$ there is at least one transvection in $G$ with residual line $L$ and fixed space $H$. Describe the automorphisms of the subgroups of $GL_2(V)$ which are rich in transvections.