I know that there is a theorem that states the following: Let $G$ be a topological group acting transitively on a connected topological space $X$. Let $x \in X$. If its stabilizer $\text{Stab}_G(x)$ is connected then so is $G$. Now I am wondering: Is there a similar theorem for simple connectedness? If both $X$ and $\text{Stab}_G(x)$ are both simply connected, what about $G$? It seems plausible to me that $G$ has to be simply connected as well in this case. However, I was not able to prove this.
Thanks in advance!