Let $k$ be a finitely generated field, $\ell$ a prime different from the characteristic of $k$, $S$ a $k$-variety, and $\mathcal{V}$ a lisse $\ell$-adic sheaf on $S$. Fix an algebraic closure $\bar{k}$ of $k$. Let $\bar{S}$ (resp. $\bar{\mathcal{V}}$) denote the pull-back of $S$ (resp. $\mathcal{V}$) to $\bar{k}$.

Every point $\bar{s} \in \bar{S}$ with image $s \in S$ gives rise to a Galois representation $\mathrm{Gal}(\bar{s}/s) \to \mathrm{GL}(\bar{\mathcal{V}}_{\bar{s}})$. Denote the connected component of the Zariski closure of the image of this representation with $G_{\bar{s}}$.

My previous question was heavily confused:

~~Are the linear algebraic groups $G_{\bar{s}}$ the geometric fibres of an affine group scheme $G \to S$?~~

~~I have some slight doubts, because $G \to S$ cannot be flat (the dimensions of the $G_{\bar{s}}$ may vary). But I think this is just because I have never worked with non-flat group schemes before, so I hope for a positive answer!~~

Of course the groups $G_{\bar{s}}$ are groups over $\mathbb{Q}_{\ell}$, and not over $\bar{s}$. So now for a more vague question:

The algebraic groups $G_{\bar{s}}$ are in some sense parameterised by $S$ (or $\bar{S}$). Is there any way to make this notion more precise? Can we express $(G_{\bar{s}})_{\bar{s}}$ as a family of algebraic groups?