Let $G$ be a Hausdorff, path-connected group acting transitively on a Hausdorff space $X$. Assume the action is continous (i.e. $(g,x) \mapsto g \cdot x$ is continuous) and transitive. It follows easily that $X$ is also path connected.
Given any path $x_t$ in $X$, must there exist a path $g_t$ in $G$ such that $x_t = g_t \cdot x_0$, all $t \in [0,1]$.
I'm inclined to think the answer is "no". In this case, I'd also be very interested to hear of additional hypotheses which make the answer into "yes". Thanks.
If $G$ is a Lie group and the stabilizer is a closed subgroup then the map $G \rightarrow X$ sending $g \mapsto gx_0$ is a principal $H$-bundle and so satisfies path lifting.
More generally, it is a theorem of Steenrod that $G \rightarrow X$ is a fiber bundle (hence, in particular, satisfies path lifting) if there exists a local section of this map around $x_0$ and the stabilizer is a closed subgroup.
I can't seem to find literature on whether or not $G \rightarrow X$ might be a fibration without actually being a fiber bundle... so I don't know. I also can't find a counterexample for when $G \rightarrow G/H$ fails to be a fibration when $H$ is closed (the trouble seems to be coming up with groups $G$ that are far from being Lie groups.)