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Write $k={\overline{\mathbb{F}}}_q$, and let $F_0$ be the entrywise $q$th-power Frobenius endomorphism of an algebraic group on the general linear group $G={\mathrm{GL}}_2(k)$. The algebraic closure is visible in the original PDF; it is lost in the converted TeX. Let $T_0$ be the diagonal maximal algebraic torus and $B_0$ the upper triangular Borel subgroup.

Choose $\beta \in k$ with $\beta -{\beta }^q=1$. Such a $\beta$ exists because $k$ is an algebraic closure, and $\beta \notin {\mathbb{F}}_q$. PutTransporting the standard ${\mathbb{F}}_q$-model through the algebraic group automorphism $\mathrm{Int}(h_{\pm })$ gives a rational structure on an algebraic group over the same field ${\mathbb{F}}_q$. In particular these are genuine Frobenius endomorphisms of an algebraic group, and their fixed-point groups areThe auxiliary $\beta$ need not be rational for the original structure: it changes the identification with the geometric group, not the field over which the transported model is defined.

For either sign, take the rational maximal torus and Borel subgroupIndeed $F_{\pm }(h_{\pm }H{h}_{\pm }^{-1})=h_{\pm }{F}_0(H)h_{\pm }^{-1}$ for $H=T_0,B_0$, so both subgroups are $F_{\pm }$-stable. Conjugacy also preserves maximality of the algebraic torus and the property of being a Borel subgroup.

Finally,Neither $c_{+}$ nor $c_{-}$ is central, so neither endomorphism is $F_0$. If $F_{+}=F_{-}$, surjectivity of $F_0$ on $G(k)$ would imply that $c_{-}^{-1}{c}_{+}$ is central. Butis not scalar in any characteristic. Thus the two nonstandard rational structures have distinct Frobenius endomorphisms, including when $q=2$.