In Proposition 3.3 of the paper "Reductive groups over finite fields" by Deligne and Lusztig, the authors mention the following fact. Let $X$ be a quasi-projective scheme over $\overline{F_p}$ and $\sigma$ is an automorphism of finite order acting on it. Both $X$ and $\sigma$ can be defined over $F_q$. Let $F$ be the corresponding geometric Frobenius. Then for $n\geq 1$, the composite $F^{n}\circ\sigma$ is the Frobenius map relative to some new way of lowering the field of definition of $X$ from $\overline{F_p}$ to $F_{q^n}$ and the Lefschetz fixed point formula for Frobenius shows that $Tr((F^{n}\sigma)^{*},H_{c}^{*}(X,Q_l))$ is the number of fixed points of $F^{n}\sigma$.
I think this means that for every $n$ there exists a twist of $X$ over $F_{q^n}$ such that when we restrict $F^{n}\sigma$ onto it we get $F^n$. I would like to know whether my understanding is correct and why such twists exist. Any comments or references will be appreciated.