Fix a field $k$. Given a discrete group $G$, the pro-algebraic completion (or Hochschild-Mostow completion) is the pro-algebraic group $A_{k}(G)$ with is universal with respect to finite dimensional representations of $G$. We usually omit the field $k$ from the notation when it is clear.

I have encountered pro-algebraic completions in a number of places, especially in relation to the schemetization problem, however there are next to no examples of pro-algebraic completions of interesting groups anywhere in the literature.

The only example that I have found is of the pro-algebraic completion of $\mathbb{Z}$, which is

$$A(\mathbb{Z})=\mathbb{G}_{a}(k)\times T\times \widehat{\mathbb{Z}},$$

where $T$ denotes denotes the pro-torus whose character group is $\mathrm{Hom}(\mathbb{Z},k^{\times})$, and $\widehat{\mathbb{Z}}$ denotes the pro-finite completion of the integers. This example, and its direct generalisation to abelian groups, was found in the following paper of Bass et.al.

Can anyone provide any other examples, or references containing examples, of pro-algebraic completions?

Tangentially, does anyone know of any methods to compute pro-algebraic completions?

I am particularly interested in understanding the pro-algebraic completions of the symmetric groups and the free groups.