On interesection of two nilpotent subgroups in small finite groups

It is proved that if $G$ is a finite group whose socle is some simple group from "Atlas of finite groups" then, for any nilpotent subgroups $A$ and $B$ of $G$, there exists an element $g$ of $G$ such that $A\cap B^g=1$, besides several cases when $A$ and $B$ are $2$- or $3$-groups.