Let $X$ be a smooth variety over a field $K \subset \mathbb{C}$. The singular fundamental group $\pi_1(X^{\operatorname{an}}, x)$ 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 $\pi_1^{\operatorname{\acute{e}t}}(X, x)$, 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 $\pi_1^{\operatorname{pro-alg}}(X_{\mathbb{C}}, x)$, defined by Simpson in Higgs bundles and local systems, which is the pro-algebraic completion of $\pi_1(X^{\operatorname{an}}, x)$. 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 $\pi_1(X^{\operatorname{an}}, x)$. 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 $X$?