Let $K$ be a number field. Let $G_K$ be its absolute Galois group. Let $p$ be a rational prime.

Let $\mathcal{R}_{K,p}^g$ be the category of finite-dimensional continuous $p$-adic representations of $G_K$ that come from geometry. Thus, an object $V$ of $RG_{K}^p$ is isomorphic to a sub-quotient of $H^i_{ét}(X, \mathbb{Q}_p)(n)$ for some smooth projective variety $X$ and integers $i$ and $n$.

$\mathcal{R}_{K,p}^g$ should be a Tannakian category, and therefore, have a Tannakian fundamental group scheme, $\mathcal{G}_{K,p}^g$, associated to it.

How is $\mathcal{G}_{K,p}^g$ related to $G_K$?