On intersection of primary subgroups of odd order in finite almost simple groups

We consider the question of the determination of subgroups $A$ and $B$ such that $A\cap B^g\ne1$ for any $g\in G$ for a finite almost simple group $G$ and its primary subgroups $A$ and $B$ of odd order. We prove that there exist only four possibilities for the ordered pair $(A,B)$.