On some subgroups of linear groups over $\mathbb{F}_2$ generated by elements of order $3$

Let $V$ be a vector space over the field of order $2$. We investigate subgroups of the linear group $GL(V)$ which are generated by a conjugacy class $D$ of elements of order $3$ such that all $d$ in $D$ have $2$-dimensional commutator space $[V,d]$.