# Is the group action of quotient group $G/H$ on quotient manifold $M/H$ proper?

My question arises from Chapter 21 of Lee's book, Introduction to Smooth Manifolds, 2nd edition.

Let $$G$$ be a Lie group acting smoothly, freely and properly on a manifold $$M$$ on the left, denoted by $$g\cdot p$$, $$g\in G, p\in M$$. Let $$H$$ be a closed normal subgroup of $$G$$. Then we have the following conclusions by some theorems in Lee's book

• By Quotient Manifold Theorem, the orbit space $$M/G$$ is a smooth manifold, such that the quotient map $$\pi_G:M \to M/G$$ is a smooth submersion;
• For the same reason, $$M/H$$ is a smooth manifold such that the quotient map $$\pi_H:M \to M/H$$ is a smooth submersion;
• By Homogeneous Space Construction Theorem, the coset space $$G/H$$ is a homogeneous $$G$$-space;
• By Quotient Theorem for Lie Groups, $$G/H$$ is also a Lie group.

By another question on topological group action, we also know that

• The group action of $$G$$ on $$M$$ induces a (left) group action of $$G/H$$ on $$M/H$$ by: $$\begin{equation}\tag{1} gH\cdot (H\cdot p) := H\cdot(g\cdot p),\quad p\in M, g\in G. \end{equation}$$
• The induced action of $$G/H$$ on $$M/H$$ is also smooth and free.
• The quotient spaces $$(M/H)/(G/H)$$ and $$M/G$$ are identical as sets, and have same quotient topology.

The natural question is, whether the induced action defined in $$(1)$$ is proper? If so, then the quotient manifold $$(M/H)/(G/H)$$ is well defined, and we can easily check that $$(M/H)/(G/H)$$ and $$M/G$$ have same smooth structure (by Quotient Manifold Theorem). But I do not know how to check this properness...

I can say more in the very special case that $$H$$ and $$G/H$$ are both compact subgroups. By Exercise 13 in Section 26 of Munkres' topology book, $$G$$ is also compact. Then by Corollary 21.6 in Lee's book, the two group actions of $$G$$ on $$M$$ and $$G/H$$ on $$M/H$$ are both proper.