I am interested in the action of field automorphism group $Aut(\mathbb{C}/\mathbb{Q})$ on
$Ext^1(\mathbb{C}^{\star},\mathbb{Z})$ or $Ext^1(\mathbb{\bar Q}^*,\mathbb{Z})$, and, more generally, on
$Ext^1_{EndA}(A(\mathbb{C}),\pi_1(A(\mathbb{C},0))/Aut_{EndA}(\pi_1(A(\mathbb{C},0) )$ for $A$
a commutative algebraic group. I am wondering whether this kind of questions have been considered
in algebraic geometry, e.g. in relation to the full *algebraic* fundamental group of
A(C) or $A(\mathbb{\bar Q})$ as a scheme over $\mathbb Q$.

These have been studied in logic (see http://arxiv.org/abs/math/0511591, http://arxiv.org/abs/0704.3561 , and also this ), and are related to questions of arithmetics, such as Galois action on a Tate module.

The main result of these papers is the following; from the point of view of logic it is a *categoricity* theorem. First note that $ exp :\mathbb{C}\rightarrow \mathbb{C}^\star$ gives an element of $Ext^1(\mathbb{C}^{\star},\mathbb{Z})$ (and also $Ext^1(\mathbb{\bar Q}^*,\mathbb{Z})$). Then the orbit of this extensions consists of all extensions where the middle is an $\mathbb{Q}$-vector space. A similar result is known about the covers of elliptic curves in char 0.
Also, I may mention that there are partial analogues in prime characteristic.