Image of the Lang-Steinberg map on disconnected centralizers of semisimple elements

Question

Let $\newcommand{\dbF}{\mathbb F}\dbF_q$ be a finite field and let $G\subseteq\mathrm{GL}_N(\bar{\dbF}_q)$ be a connected reductive group defined over $\dbF_q$. Let $F$ be the associated Frobenius map, such that $G(\dbF_q)=G^F$.

Let $g\in G$ be a semisimple $F$-fixed element such that $C_G(g)$ is disconnects. The Lang-Steinberg Theorem implies that the restriction of the map $L(x)=x^{-1}F(x)$ to $C_G(g)^\circ$ (the identity component) is onto $C_G(g)^\circ$, and hence $L({C_G(g)})\supseteq C_G(g)^\circ$.

Is anyone aware of an example where this inclusion is proper?

Clearly, this inclusion is an equality if and only if all cosets of $C_G(g)/C_G(g)^\circ$ contain an $F$-fixed point, for then given such a coset $A$ we may write $A=xC_{G}(g)^\circ$ for $x$ such that $F(x)=x$, and then, given $y=xz\in A$ with $z\in C_{G}(g)^\circ$, $L(y)=L(z)\in C_G(g)^\circ$. Also, to show that $A^F\ne \emptyset$ it suffices to show $F(A)=A$.

So, to answer my question it would suffice to find a $g\in G^F$ semisimple for which some of the cosets of $C_G(g)$ are not $F$-stable. So far I haven't managed to find such an example, or prove the one cannot exist. Would appreciate any help on the subject greatly!

Caveat I'm mainly interested in things that occur for almost all ground field characteristics. Examples in small characteristics are undoubtably interesting, but I would mostly like to know whether this kind of phenomenon can occur over unboundedly many prime characteristics.

Answer 1

One example is $G = \operatorname{PGL}_4$, $g$ a semisimple lift of the Coxeter element of the Weyl group, say as $g = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix}$, and $q \equiv 3 \pmod4$. The component containing $\operatorname{diag}(1, \zeta_4, -1, -\zeta_4)$ is not defined over $\mathbb F_q$.