Let $H$ be an infinite subgroup of a discrete group $G$. $H$ is called nearly normal if it is commensurable with a normal subgroup $K$ of $G$, that is $H\cap K$ is a finite index subgroup of both $H$ and $K$. If $H$ contains a finite index subgroup which is also a normal subgroup of $G$, then $H$ is obviously nearly normal. Is the converse also true? Or does there exist an example of a nearly normal subgroup $H$ which does not contain any finite index subgroup which is normal in $G$? If there is such a counterexample, can we at least show that every nearly normal subgroup contains a non-trivial normal subgroup of $G$? Or perhaps there is a counterexample for this question too?
Let $K$ be an infinite product of cyclic groups of order 2, and $H$ an index two subgroup of $K$. Let $A$ be the full automorphism group of $K$, and $G=K\rtimes A$.
Then since $G$ acts transitively on the non-trivial elements of $K$, $H$ has no non-trivial subgroup normal in $G$.