Let $G$ be an infinite group that acts transitively on an infinite set X. If $H$ is a subgroup of $G$ of finite index, does this imply there are only finitely many $H$-orbits when $H$ acts on $X$?
A professor of mine made a remark with the following specifications: We had $G=SL_2(\mathbb{Z})$, $X=\mathbb{Q}\cup\{\infty\}$, and $H$ a congruence subgroup of $SL_2(\mathbb{Z})$. I don't know why its true in this case either. Can someone shed some light?