Is a finite group a subgroup of the automorphism group of a variety?

Question

Let $G$ be a finite group with $n$ elements acting on $k[x_1,\cdots,x_n]$ through the regular representation. Is it true that $G$ is a subgroup (in the sense that there exists an injective group homomorphism) of the automorphism group $Aut(V)$ of the variety $V:=k^n/G$? If so, where can I read a proof for this? Thanks for your help!

Answer 1

The answer to the question in your title is yes. This follows from the fact that every finite group $G$ is a Galois group of a finite Galois covering $X\rightarrow\mathbb{P}^1_{\mathbb{C}}$, and hence $G\subset\text{Aut}_{\mathbb{C}}(X)$.