Example of a non normal subgroup of finite index in an infinite group

Question

I know that if an infinite group $G$ that contains a subgroup $H$ of finite index, then $H$ contains a subgroup $K$ of finite index which is normal in $G$.

But are there examples where $H$ is a subgroup of $G$ finite index and $G$ is infinite yet $H$ is not normal?

Answer 1

Consider the group $G=\mathbb{Z}\times S_3$ and its subgroup $H=\mathbb{Z}\times \langle(12)\rangle$. We have $|G:H|=3$ and $H$ is not normal in $G$.