Grothendieck in SGA 1 introduces a proposition in expose 5 (proposition 3.1) which states:

Let $X$ be etale, separated of finite type over $Y$, locally noetherian, and let $G$ be a finite group which operates on $X$ by $Y$-automorphisms.  Then $G$ operates admissibly and the quotient scheme $X/G$ is etale over $Y$.

The hint he gives is that we may show this for $X$ quasi-projective, and to use proposition 1.8, which states that $G$ operates admissibly on $X$ iff $X$ is the union of open affines that are invariant under the action of $G$.

I am unsure how to show this.  Help please?