Let $G$ be a finite group acting on $\mathbb A^n\_{\mathbb C}$. Let $Y$ be a dense open whose complement is of codimension at least two. 


Assume $Y$ is $G$-stable, the action of $G$ is free on $Y$, and that $Y/G$ is a quasi-affine scheme.



> 
> Is the action of $G$ on $\mathbb A^n\_{\mathbb C}$ free? 
> 
> 
> 



Group actions on affine space which are almost good
The answer in general is **no**, as shown by the following example.


Take the group $G=\mathbb{Z}/2 \mathbb{Z}$, acting on $\mathbb{A}^2$ as $$(x, \, y) \mapsto (-x, \, -y).$$ This action is free outside the unique fixed point $p=(0, \, 0)$, so $Y= \mathbb{A}^2-\{p\}$ is $G$-stable. 


Furthermore, we have $$\mathbb{A}^2/G = \textrm{Spec} \, \mathbb{C}[x, \, y]^G=\textrm{Spec} \, \mathbb{C}[x^2, \, xy, \, y^2] = \textrm{Spec} \, \mathbb{C}[u, \, v, \, w]/(v^2-uw).$$
So $\mathbb{A}^2/G$ is an affine variety isomorphic to a quadric cone in $\mathbb{A}^3$, whereas $Y/G$ is this cone minus its vertex. 


