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 $L({C\_G(g)})\supseteq C\_G(g)^\circ$. 



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



Image of the Lang-Steinberg map on disconnected centralizers of semisimple elements
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 $\mathbb F\_q$.


