Orbit space of action of a subgroup of a Lie group on a separable metric space

Question

I am stuck on this question. Let $G$ be a Lie group acting freely on a separable metric space $X$. Assume that the orbit space $X/G$ is Hausdorff. Let $H$ be a normal Lie subgroup of $G$. Is the orbit space $X/H$ Hausdorff?

If this is not the case, what if the quotient $G/H$ is compact?

Thank you