Assume that $\Psi: G \times Y \to Y$ is a topological group action (so $\Psi$ is continuous). Let $[\cdot]: Y \to Y_{/G}$ be the continuous projection map, sending each $y\in Y$ to its orbit. Also, assume $s: Y_{/G} \to Y$ is a continuous section: $$\forall [y] \in Y_{/G}: \left[s\left(\left[y\right]\right)\right]=[y]$$
Moreover, assume that $G$ acts transitively and faithfully on $[]^{-1}(y)$, for any $y \in Y$.
Can I assume that the mapping $$Y \to G, ~~ y \mapsto g_y$$ where $g_y$ is such that $$g_y\cdot s([y])=y$$ is continuous?
Thank you for your help!