On intersections of abelian and nilpotent subgroups in finite groups

Let $A$ be an abelian subgroup of a finite group $G$, and let $B$ be a nilpotent subgroup of $G$. If $G$ is solvable, then we prove that it contains an element $g$ such that $A\bigcap B^g\le F(G)$, where $F(G)$ is the Fitting subgroup of $G$. If $G$ is not solvable, we prove that a counterexample of smallest order to the conjecture that $A\bigcap B^g\le F(G)$ for some element $g$ of $G$ is an almost simple group.