Suppose $F$ is a function space $Y^X$ with $Y$ discrete (so it has the topology of pointwise convergence), and $F'$ is another function space $Y'^{X'}$ with $Y'$ discrete, and suppose we have an action of some group $G$ on $X'$. We can extend this action to an action on $F'$ via $g.f' := (x' \mapsto f'(g.x'))$. Let $\pi : F' \to F'/G$ be the quotient map of this action.
Is it true that any continuous map $\theta : F \to F'/G$ factors through $\pi$? In other words, is it true that there is some continuous map $\tilde\theta : F \to F'$ for which $\theta = \pi \circ \tilde\theta$? Equivalently, is there a continuous choice of representatives $r_f \in \theta(f)$ for $f \in F$?
I am not even sure about the case when $G$ acts regularly on $X'$ (i.e. $X'$ is a $G$-torsor), though I expect the answer to be "yes" in this case.