@user125763: trivial residue field extension at the marked points (so ensures induced map on completions from the etale morphism is an isomorphism).

The same issue comes up when describing families of semistable curves (say with smooth generic fiber): one can always get by using the etale topology (appropriately formulated) without recourse to analytic methods, ultimately due to the Artin approximation theorem.

@user125763: It also follows from Artin approximation (the earlier paper of Artin...) that if $(X,x)$ and $(X',x')$ are pointed schemes of finite type over a field or excellent Dedekind domain $R$ and $f:\mathscr{O}_{X,x}^{\wedge} \simeq \mathscr{O}_{X',x'}^{\wedge}$ is an $R$-isomorphism then there is a common residually trivial pointed etale neighborhood $(X'',x'')$ of $(X,x)$ and $(X',x')$ which induces an isomorphism between those completed local rings that agrees with $f$ modulo whatever power of the maximal ideals you like. Same for arbitrary excellent schemes via Popescu's approx. thm.

Why do you believe that $X \times_Y \widetilde{Y} \rightarrow \widetilde{Y}$ is the quotient by the natural $G$-action on the source (say assuming $X$ and $Y$ are quasi-projective varieties over an algebraically closed field)? This entails some delicate algebraic conditions, not just topological ones, and the formation of such quotients generally does not commute with non-flat base change (beyond the case of a free $G$-action).

In your comment above I think you meant to write "split" rather than "quasi-split", so it remains unclear what you mean by "quasi-split" (do you follow SGA3 or have another definition in mind)? Maybe some context and motivation would help.