On intersection of primary subgroups in the group $\mathrm {Aut(F_4(2))}$

It is proved that, in a finite group $G$ which is isomorphic to the group of automorphisms of the Chevalley group $F_4(2)$, there are only three possibilities for ordered pairs of primary subgroups $A$ and $B$ with condition: $A\cap B^g\ne 1$ for any $g\in G$. We describe all ordered pairs $(A,B)$ of such subgroups up to conjugacy in the group $G$ and in particular, we prove that $A$ and $B$ are $2$-groups.