On intersections of primary subgroups in the group Aut$(L_n(2))$

It is proved that, in a finite group $G$ whose socle is isomorphic $L_n(2)$, there exist primary subgroups $A$ and $B$ such that the intersection of $A$ and any subgroup conjugate to $B$ under the action of $G$ is nontrivial only if $G$ is isomorphic to the group Aut$(L_n(2))$; in this case, $A$ and $B$ are 2-subgroups. All ordered pairs $(A,B)$ of such subgroups are described.