Tits has proved that a finite simple group $G$ with a BN-pair of rank $n \ge 3$, is of Lie type. Let $B$ be the Borel subgroup and $(W,S)$ the Coxeter system. The subset lattice of the set $S$ is lattice-equivalent to the interval $[B,G]$ of parabolic subgroups, so is equivalent to the boolean lattice of rank $n$.
We ask about the following extension of Tits' theorem:
Is a finite simple group $G$ having a subgroup $H$ with $[H,G]$ boolean of rank $n \ge 3$, of Lie type?
For $|G| < 4 \cdot 10^6$ (using GAP) the only examples are ${\rm A}_3(2)$, $^2{\rm A}_2(5^2)$, ${\rm C}_3(2)$ and $^2{\rm A}_3(3^2)$.
