On integral images of Curtis homomorphisms for ${\mathrm{G}\mathrm{L}}_n$ and ${\mathrm{U}}_n$

For $G={\mathrm{G}\mathrm{L}}_n$ or ${\mathrm{U}}_n$ defined over a finite field of characteristic $p$, we refine a result of Bonnafé and Kessar on the saturatedness of the Curtis homomorphism ${\mathrm{C}\mathrm{u}\mathrm{r}}^G$ by describing the image of ${\mathrm{C}\mathrm{u}\mathrm{r}}^G$ over $\bar{\mathbb{Z}}[1\mathrm{/}p]$ via a system of linear conditions.