  For $G = \mathrm{GL}_n$ or $\mathrm{U}_n$ defined over a finite field of
characteristic $p$, we refine a result of Bonnaf\'e and Kessar on the
saturatedness of the Curtis homomorphism $\mathrm{Cur}^G$ by describing the
image of $\mathrm{Cur}^G$ over $\overline{\mathbb{Z}}[1/p]$ via a system of
linear conditions.
