Galois action on the pro-algebraic completion of the singular fundamental group

Question

Let $X$ be a smooth variety over a field $K \subset \mathbb{C}$. The singular fundamental group $\pi_1(X^{\text{an}}, x)$ generally does not carry an action of the absolute Galois group $\operatorname{Gal}(\overline{K}/K)$. However, its profinite completion yields the étale fundamental group $\pi_1^{\text{ét}}(X, x)$, which has a natural Galois action due to its algebraic definition.

When $X$ is projective, there are other fundamental groups that are larger than the étale fundamental group but smaller than the singular fundamental group.

1. The pro-algebraic fundamental group $\pi_1^{\text{pro-alg}}(X_{\mathbb{C}}, x)$, defined by Simpson in Higgs bundles and local systems, which is the pro-algebraic completion of $\pi_1(X^{\text{an}}, x)$. Since it is defined algebraically, it has a Galois action (though I am not entirely sure about this point).
2. The $S$-fundamental group $\pi_1^S(X, x)$ defined by Langer, which also carries a Galois action due to its algebraic construction.

These groups are defined for projective varieties. However, for non-proper $X$, we can still consider the pro-algebraic completion of $\pi_1(X^{\text{an}}, x)$. This raises the following questions.

1. Can we define a natural Galois action on the pro-algebraic completion of the singular fundamental group for non-proper $X$?
2. Can this pro-algebraic completion be defined algebraically for non-proper $X$?

Answer 1

You can define $\pi_1^\text{pro-alg}(X,x)$ with no properness assumption as the Tannakian group associated to the Tannakian category of algebraic vector bundles with flat connection, regular at infinity (with fiber functor given by the fiber at $x$). Over $\mathbb{C}$, it is always isomorphic to the pro-algebraic completion of the topological fundamental group. See Deligne, Le Groupe Fondamental de la Droite Projective Moins Trois Points, Proposition 10.32.

It is clear from the algebraic nature of the definition that the group of $\overline{K}$-points of $\pi_1^\text{pro-alg}(X,x)$ and the group of $\overline{K}$-points of $\pi_1^\text{pro-alg}(X_{\overline{K}},x)$ both carry an action of $\operatorname{Gal}(\overline{K}/K)$ (these are two different groups because $\pi_1^\text{pro-alg}$ is not compatible with base change!). If you replace $\overline{K}$ with $\mathbb{C}$ in the above, then you get an action of the group of automorphisms of $\mathbb{C}$ over $K$ but I don't think you have a natural action of $\operatorname{Gal}(\overline{K}/K)$.

A slightly better-behaved variant (notably, compatible with base change) is the de Rham fundamental group $\pi_1^\text{DR}(X,x)$, which is the Tannakian group of nilpotent algebraic vector bundles with flat connection, regular at infinity. Over $\mathbb{C}$, this is the pro-unipotent completion of the topological fundamental group.