If $G$ is a virtually abelian group and $H$ is a finite index subgroup of $G$. Is it always true that $H$ is virtually abelian ?
Since $G$ is V.A, it has a finite index subgroup $K$ which is abelian.
If $H \subset K$, then $H$ is abelian and therefore V.A.
If $K \subset H$, then $H$ is V.A.
What about when neither subgroup is contained in the other ?