Intersections of primary subgroups in nonsoluble finite groups isomorphic to $L_n(2^m)$

Given a finite group $G$ with socle isomorphic to $L_n(2^m)$, we describe (up to conjugacy) all ordered pairs of primary subgroups $A$ and $B$ in $G$ such that $A\cap B^g\ne1$ for all $g\in G$.