As this [question](http://mathoverflow.net/questions/81146/finite-index-normal-subgroup-in-a-normalised-union-of-two-groups-closed) was closed as a duplicate of [Existence of simultaneously normal finite index subgroups](http://mathoverflow.net/questions/34592/existence-of-simultaneously-normal-finite-index-subgroups), I am opening a modified version here.

$G$ is a group, $A$ and $B$ are two subgroups of $G$. Suppose that $A∩B$ has finite index in both $A$ and $B$. It has been shown [here](http://mathoverflow.net/questions/34592/existence-of-simultaneously-normal-finite-index-subgroups) that $A\cap B$ need not have a subgroup of finite index which is normal in both $A$ and $B$.

**Question** : If the set $A\cup B$ is normalised by $G$, does $A\cap B$ has a subgroup of finite index which is normal in both $A$ and $B$.