Existence of simultaneously normal finite index subgroups
$G$ is a group, $A$ and $B$ are two subgroups of $G$. Let $H$ be a subgroup of $A\cap B$ which is of finite index in both $A$ and $B$. Does $H$ has a subgroup of finite index which is normal in both $A$ and $B$?
Edit : In view of the negative answer pointed below, let me add the additional requirement that the union $A\cup B$ be normalised by $G$.