 Rational structure on an algebraic group

A rational structure over ${\mathbb{F}}_q$ is a model over that finite field together with its identification after extension to ${\overline{\mathbb{F}}}_q$. It determines a Frobenius endomorphism of an algebraic group, whose fixed points are the rational points. Conjugating a standard Frobenius by an algebraic group automorphism transports the rational structure, even if the automorphism is not itself rational for the original structure.