Intersections of two nilpotent subgroups in finite groups with socle $L_2(q)$

Given a finite group $G$ with socle isomorphic to $L_2(q)$, $q\ge4$, we describe, up to conjugacy, all pairs of nilpotent subgroups $A$ and $B$ of $G$ such that $A\cap B^g\ne1$ for all $g\in G$.