I am a little bit confused about global quotient orbifolds. I dont know if there are any conditions that must be satisfied by the group action on the manifold. This is what I thought could be done:
Let $M$ be a smooth $n$-manifold an $G$ a finite group acting on it smoothly. Let us put an obifold structure on $X:=M/G$. A global uniformizing system for $X$ is given by $(M,G,\pi)$ where $\pi\colon M\rightarrow M/G$ is the natural projection map.
Now we can make uniformizing systems for neighborhoods of points in $X$. For any point $x\in X$, choose a connected open neighborhood of $x$. Choose a connected component of $\pi^{-1}(U_x)$, say $\tilde{U}_x$, and let $\tilde{G}_x$ be the subgroup of $G$ such that $\tilde{G}_x\cdot\tilde{U}_x\subset \tilde{U}_x$. Then $(\tilde{U}_x,\tilde{G}_x,\pi|_{\tilde{U}_x})$ is a uniformizing system for $U_x$.
My question is whether this description is complete.