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.