Fundamental groups of coset spaces

Question

Let $G,H$ be topological groups. In the case that $G$ is connected and simply connected, we have $$\pi_1\left(G/H\right)\simeq\pi_0\left(H\right)\simeq H/H_0$$ where $H_0\leq H$ is the component of $H$ connected to identity.

Now suppose $G$ is not simply connected. In a paper 1, I have found the following construction. Let $p\colon\tilde G\to G$ be the universal cover. Then $\pi_1\left(G/H\right)\simeq\pi_1\left(\tilde G/\tilde H\right)$ where $\tilde H=p^{-1}\left(H\right)$. However I can't seem to prove this. Is it true? I am attaching the paragraph where this claim is made (From section 5.1.1)

1: Maurice Kléman, Curved crystals, defects and disorder, Advances in Physics (Volume 38 Issue 6), 1989

Answer 1

By construction $p$ induces a bijective identification of groups: $$p\colon\tilde{G}/\tilde{H}\to G_0/(G_0\cap H)\cong (G/H)_0$$ That is, by definition of $G_0$ we may pick a preimage of any of its elements in $\tilde{G}$, so the map is surjective. On the other hand it is injective by definition of $\tilde{H}$. Thus: $$\pi_1(\tilde{G}/\tilde{H})\cong \pi_1((G/ H)_0)\cong \pi_1(G/H)$$