  Let $\mathbf{G}$ be a connected reductive group with connected center defined
over $\mathbb{F}_q$, with Frobenius morphism F. Given an irreducible complex
character $\chi$ of $\mathbf{G}^F$ with its Jordan decomposition, and a Galois
automorphism $\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, we
give the Jordan decomposition of the image ${^\sigma \chi}$ of $\chi$ under the
action of $\sigma$ on its character values.
