I have been given the following problem:
Problem. Let $G$ be a group and $H$ a subgroup. Assume that $G$ acts on $X$ and that $H$ acts transitively on $X$. Determine whether it is true or not the following statement: $G=G_\omega H$ for all $\omega\in X$, where $G_\omega$ denotes the stabilizer of $\omega$ in $G$.
Thinking about it, I have seen that if the conclusion is true, we would have $$|G|=|G_\omega||H|/ |G_\omega \cap H|=|G_\omega||H|/|H_\omega|.$$ Now, $H$ acts transitively, so $|H|/|H_\omega|=|X|$, and thus $|G:G_\omega|=|X|$, which implies that the orbit of $\omega$ in $G$ is $X$, and thus $G$ acts also transitively.
Am I in the good direction?
You attempt to argue only with cardinalities - which won't help in the infinite case. (Of course, your conclusion that $G$ acts transitively on $X$ is true whatsoever as already the subgroup $H$ acts transitively).
Let $\omega \in X$. Our mission is to find, for any $g\in G$, elements $\gamma\in G_\omega$ and $h\in H$ with $g=g_\omega h$; in other words, we want $gh^{-1}\in G_\omega$ . Let $\alpha=g^{-1}\cdot \omega$. By transitivity of $H$, there exists $h\in H$ with $h\cdot \alpha=\omega$. Then $h^{-1}\cdot \omega=\alpha$ and hence $gh^{–1}\cdot \omega=\omega$. In other words, $gh^{-1}\in G_\omega$ as desired.