I am following the proof of Lemma 2.11 in Moerdijk, Mircun, "Introduction to Foliations and Lie groupoids". There is a claim I don't understand: suppose $V$ is an open set in a smooth manifold $M$ and have a smooth function $f: V \rightarrow M$, such that for some finite subgroup $G \subset \operatorname{Diff}(M)$, $f(x) \in G \cdot x$ for any $x \in V$. Then on any connected component of the intersection of $V$ with the set of points which have no stabilizer under $G \setminus \{ \operatorname{id} \}$, $f$ must be the action of an element in $G \setminus \{ \operatorname{id} \}$ on that component.
From the way this claim is phrased, it makes me think that it is a special case of the more general claim:
Let $G$ be a finite group and $\pi: E \rightarrow B$ be a principal $G$-bundle where $E$ is connected. Suppose $f: E \rightarrow E$ is a continuous map satisfying $\pi \circ f = \pi$. Then $f$ is just the action of a $g \in G$. However, in this generality the claim seems false; for example we can just take the map $f$ defined on fibers as $f(h) = e$ where $e$ is the neutral element of $G$.