Some Notions
A variety over a field is defined to be a scheme of finite type over this field.
An $\mathbb{F}_q$-rational structure of an $\bar{\mathbb{F}}_q$-variety $V$ is a $\mathbb{F}_q$-variety $V_0$ with an isomorphism $V\cong V_0\times_{\mathbb{F}_q}\bar{\mathbb{F}}_q$ of varieties over $\bar{\mathbb{F}}_q$.
Along with the above notations, let $\mathrm{Frob}$ be the absolute Frobenius endomorphism of $V_0$, and we call $F:=\mathrm{Frob}\otimes \mathrm{ Id_{\bar{\mathbb{F}}_q}}$ the geometric Frobenius endomorphism of $V$ associated to the rational structure given by $V_0$. It is clear that geometric Frobenius endomorphisms and rational structures are $1-1$ corresponded.
Along with the above notations, we call a subvariety $W$ of $V$ is rational if $FW=W$. In this case one can show $W$ also admits a rational structure by the restriction of $F$.
More infomation about these terminologies can be found in Digne & Michel Representations of Finite Groups of Lie type, and Malle & Testerman Linear Algebraic Groups and Finite Groups of Lie Type.
An Example
Different geometric Frobenius morphisms can be very different. Consider the general linear group $GL_n(\bar{\mathbb{F}}_q)$, then it has a natural geometric Frobenius $F$ (the one taking each entry of a matrix to its $q$-th power) with fixed points $$GL_n(\bar{\mathbb{F}}_q)^F=GL_n(\mathbb{F}_q).$$ Meanwhile, $F'(-):=(F(-)^T)^{-1}$, where $T$ means taking transpose, defines another geometric Frobenius (hence another rational structure over $\mathbb{F}_q$), and it has fixed points $$GL_n(\bar{\mathbb{F}}_q)^{F'}=U_n(\mathbb{F}_q)$$ the unitary group over $\mathbb{F}_q$.
My Question
From now on let $G$ be a connected reductive group over $\bar{\mathbb{F}}_q$, with a rational structure over $\mathbb{F}_q$, denote by $F$ the associated geometric Frobenius endomorphism, and let $B$ be a rational Borel subgroup. Then $F$ acts on the closed points of the flag variety $X=G/B$ over $\bar{\mathbb{F}}_q$.
My Question: Does the action of $F$ on $X$ give an $\mathbb{F}_q$-rational structure on $X$, i.e. does $F$ act on $X$ as a geometric Frobenius with respect to some rational structure?
Any references or suggestions are appreciated.
Some Ideas
I'm not familiar with the construction of quotients, but I am trying to split the question into the following (possibly vague) steps, please ignore them if they are just don't make much sense:
Can the action of $F$ on $X$ be realized as a morphism of schemes (Carnahan confirmed $F$ is a morphism in the below comment, thanks)?
Let $V_0$ be the rational structure of $G$ associated to $F$, and let $W_0$ be the ratoinal structure of $B$, must $V_0$ be a group scheme over $\mathbb{F}_q$, is it always the case that $W_0$ be a closed subscheme of $V_0$?
Does the quotient scheme $V_0/W_0$ exist?
Should we have that $V_0/W_0\times_{\mathbb{F}_q}\bar{\mathbb{F}}_q\cong G/B$, and $F=\mathrm{Frob}_{V_0/W_0}\otimes \mathrm{Id}_{\bar{\mathbb{F}}_q}$?