It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference or two to help me get some things straightened out.

One can construct a category $\mathcal{M}$ of 0-dimensional motives over $\mathbb{Q}$. If I'm not mistaken, one possible construction is to start with the category of 0-dimensional varieties over $\mathbb{Q}$, with correspondences as morphisms, and formally adjoin images of idempotents. This $\mathcal{M}$ is a neutral Tannakian category, meaning it is equivalent to the representations of some pro-algebraic group. In this case that group should be the absolute Galois group of $\mathbb{Q}$.

There are several fiber functors $\omega_{dR},\omega_{\ell},\ldots$ from $\mathcal{M}$ to vector spaces (over various fields), given by the different motivic realizations (de Rham, $\ell$-adic,...). Each fiber functor $\omega_\bullet$ has an automorphism group $G_{\mathbb{Q},\bullet}$, which is an affine pro-algebraic group. As I understand it, the groups $G_{\mathbb{Q},\bullet}$ should be viewed as realizations of some motivic absolute Galois group $G_{\mathbb{Q}}$. Each functor $\omega_\bullet$ identifies $\mathcal{M}$ with the representations of $G_{\mathbb{Q},\bullet}$.

What are the groups $G_{\mathbb{Q},\bullet}$, as $\bullet$ ranges over the different realizations on $\mathcal{M}$? Can someone point me to a reference where these groups are described?

I would love to find a reference that also describes the construction of $\mathcal{M}$.