Let $G$ be a group, $S$ a set and $\chi$ an action of $G$ on $S$. This defines a functor $F$ from the group as a single object category to the category of sets. Let $\phi$ be an automorphism of $G$, which can also be viewed as a functor from $G$ to $G$. I'm interested in determining what are the requirements for the existence of a natural transformation from $F$ to $F \circ \phi$.
Here is what I've determined so far. If $\chi$ is a simply transitive action, then there always exist such a natural transformation, which induces a map $G \to G: g \to \phi(g).g_0$, where the choice of $g_0 \in G$ is free.
If $G$ has a normal subgroup $N$ which acts simply transitively on $S$, then the natural transformation, if it exists, also induces a similar map. However, some conditions must be filled on $g_0$. I do not know if there is a general answer in this case. More generally, is there a general solution for any $G$ ?
Edit: Someone suggested to me recently that this problem induces diagrams which might be similar to Kan extensions. I have checked the definition but I'm unable to concretely build such an extension. Can it be that Kan extensions may help in this case ?