= Solution
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 $\beta\notin\mathbb F_q$. Put
$$
h_+=\begin{pmatrix}1&\beta\\0&1\end{pmatrix},\qquad h_-=\begin{pmatrix}1&0\\\beta&1\end{pmatrix},\qquad F_\pm=\operatorname{Int}(h_\pm)\circ F_0\circ\operatorname{Int}(h_\pm)^{-1}.
$$
Transporting the standard $\mathbb F_q$-model through the <algebraic group> automorphism $\operatorname{Int}(h_\pm)$ 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 are
$$
G^{F_\pm}=h_\pm\mathrm{GL}_2(\mathbb F_q)h_\pm^{-1}.
$$
The 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 subgroup>
$$
\boxed{T_\pm=h_\pm T_0h_\pm^{-1}\ \subset\ B_\pm=h_\pm B_0h_\pm^{-1}.}
$$
Indeed $F_\pm(h_\pm Hh_\pm^{-1})=h_\pm F_0(H)h_\pm^{-1}$ 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,
$$
F_\pm=\operatorname{Int}(c_\pm)F_0,\qquad c_+=\begin{pmatrix}1&1\\0&1\end{pmatrix},\quad c_-=\begin{pmatrix}1&0\\1&1\end{pmatrix}.
$$
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. But
$$
c_-^{-1}c_+=\begin{pmatrix}1&1\\-1&0\end{pmatrix}
$$
is not scalar in any characteristic. Thus \b[the two nonstandard rational structures have distinct Frobenius endomorphisms], including when $q=2$.
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