Let $G$ be a finite étale group scheme over a field $k$ and $X=\rm{Spec}(A)$ be an affine scheme on which $G$ acts. The categorical quotient $X/G$ exists and may be described as $\rm{Spec}(A^H)$ where $H$ is the Hopf algebra associated to $G$ and $A^H$ is the coequalizer of the coaction of $H$ on $A$ and $a\mapsto a\otimes 1$.

Nevertheless, I feel uncomfortable with this description all the more so as $G$ is a finite étale group scheme and is well understood using the equivalence of categories between finite étale group schemes and finite $\rm{Gal}(\overline{k}/k)$-groups.

Is there a way to use the latter equivalence in order to describe the quotient? I am especially interested in the case where $X$ is a closed subscheme of the affine space.