I am encountering the following situation which is similar to the Abhyankar's
higher dimensional conjecture on étale fundamental groups, but with much
stronger assumptions:


Let $S$ be a finitely generated subring of $\mathbb{C}$, let $X$ be
a smooth affine variety over $S$, and let $G$ be a finite group such that the
following holds. For any large enough prime $p$ and a base change $S\to k$
to an algebraically closed field of characteristic $p$, the variety $X\_{k}$
admits a Galois covering with the Galois group $G$. Does this imply that
$X\_{\mathbb{C}}$ also admits a Galois covering with the group $G?$


I believe the answer is "yes" if $X$ is a curve, or is a complement of
divisors with normal crossings in a projective space (by a result of Abhyankar). 


Any suggestions or references would be greatly appreciated.



A weak version of high dimensional Abhyankar's conjecture
Here's a sketch of a possible way: Since $p$ is large enough, we can assume that $G$ is of order prime to $p$. If $X$ was projective, then the prime-to-$p$ completions of $\pi\_1(X\_{\mathbb{C}})$ and $\pi\_1(X\_{\bar k})$ would be isomorphic by the results of SGA1, and we would be done. To "reduce" to this case, we might assume that we have a smooth compactification $\bar X$ of $X$ such that $\bar X\setminus X$ is a divisor with simple normal crossings. In this case we can use Abhyankar's lemma to extend the covering of $X\_{\bar k}$ to a tamely ramified covering of $\bar X\_{\bar k}$. The magic of log geometry should allow one to lift this covering to characteristic zero exactly as in the case without ramification. In particular, a *single* $p$ (satisfying some good reduction hypotheses) should be enough.


**EDIT.** Let me add some details. First, a definition (the terminology is not standard). 



> 
> **Definition.** (1) (cf. SGA1 Exp. XIII, 2.1) Let $\pi:X\to S$ be a smooth morphism of schemes, $D\subseteq X$ an effective Cartier divisor on $X$. We say that $D$ has *normal crossings relative to $S$* if locally on $X$, there exists an etale $S$-morphism $g:X\to \mathbf{A}^n\_S$ such that $D=g^\*(D(x\_1\cdot\ldots\cdot x\_r))$ for some $r$. We will call $(X, D)$ a *log smooth pair* over $S$.
> 
> 
> (2) We say that a finite morphism $f:Y\to X$ over $S$ is a *tame cover* of a log smooth pair $(X, D)$ if etale locally on $Y$, there exists a morphism $g:X\to \mathbf{A}^n\_S$ as in (1), integers $e\_1 ,\ldots, e\_r$ invertible on $S$, and an isomorphism over $X$ between $Y$ and the pullback of $(x\_1, \ldots, x\_n)\mapsto (x\_1^{e\_1}, \ldots, x\_r^{e\_r}, x\_{r+1}, \ldots, x\_n):\mathbf{A}^n\_S\to \mathbf{A}^n\_S$ along $g$.
> 
> 
> 


Then Abhyankar's lemma (see SGA1 Exp. XIII, Appendice I, Proposition 5.5) can be phrased as follows:



> 
> **Lemma 1.** Assume that $S={\rm Spec}\, k$, and let $(X, D)$ be a log smooth pair over $S$. Set $U=X\setminus D$. Then the restriction functor
>  $$ \left( \text{tame covers of }(X,D)\right) \longrightarrow \binom{\text{finite etale covers of }U}{\text{tame along }D} $$
>  is an equivalence.
> 
> 
> 


The "log magic" I referred to above is hidden in the proof of the following lemma:



> 
> **Lemma 2.** Let $S\to \widetilde{S}$ be a nilpotent closed immersion, let $(\widetilde X, \widetilde D)$ be a log smooth pair over $\widetilde S$, $(X, D)$ its base change to $S$. Then the restriction functor
>  $$ \left(\text{tame covers of }(\widetilde X, \widetilde D)\right) \longrightarrow\left(\text{tame covers of }(X, D)\right)$$
>  is an equivalence.
> 
> 
> 


This can be deduced from Kato's first article on log geometry "Logarithmic structures of Fontaine-Illusie", section 3. (Probably there is an easier way of seeing this directly.) The point is that "tame covers" are log etale if $X$ and $Y$ are given their natural log structures. If we replace $\Omega^1\_{Y/X}$ with its variant with log poles, we get the zero sheaf, which is why we obtain a unique extension property for tame covers. 


From this we obtain the result you need. More precisely, suppose that $R$ is a henselian dvr with residue field $k$ and that $(X, D)$ is a log smooth pair over $S={\rm Spec}\, R$ such that $X$ is proper over $S$. Let $U=X\setminus D$. If $V\_k\to U\_k$ is a connected $G$-torsor, where $G$ is of order invertible in $k$, then Lemma 1 provides an extension to a connected $G$-tame cover $Y\_k$ of $(X\_k, D\_k)$. Applying Lemma 2 to all ${\rm Spec}\,k\to {\rm Spec}\, R/m^n$, we obtain a connected $G$-tame cover $\mathcal{Y}$ of the formal completion $(\mathcal{X}, \mathcal{D})$. By Grothendieck's existence theorem applied to $f\_\*\mathcal{O}\_{\mathcal{Y}}$, this comes from a connected $G$-tame cover $Y$ of $(X, D)$, which stays connected on the geometric generic fiber $(X\_{\bar K}, D\_{\bar K})$. Restricting to $U\_{\bar K}$, we obtain a desired $G$-cover of your affine scheme $U\_{\bar K}$. 


