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.