On intersection two nilpotent subgroups in small groups

In the paper we prove that if $G$ is a finite almost simple group with socle isomorphic to $G_2(3)$, $G_2(4)$, $F_4(2)$, ${}^2E_6(2)$, $Sz(8)$, then for every nilpotent subgroups $A,B$ of $G$ there exists an element $g\in G$ such that $A\cap B^g=1$, except the case $G=Aut(F_4(2))$, and $A,B$ are $2$-groups.