= 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.
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