Coarse moduli of motivic augmentations

A variety $X$ over a suitable base $Z$ gives rise to a highly structured algebra $C^*(X)$ in motives over $Z$. In turn, a $Z$-point gives rise to an augmentation $C^*(X) \to 1$. This assignment $X(Z) \to Aug(C^*(X))$ factors the so-called ``unipotent Kummer map'' to torsors under the unipotent funda