Let $p$ be a prime number, and $k$ a finite field with $q=p^n$ elements. Let $X$ be a scheme over $k$ and denote by $X'$ its base change to an algebraic closure $\bar k$ of $k$. Denote by $F_X:X\rightarrow X$ the Frobenius endomorphism of $X$ relative to $k$, and by $F'_X$ its base change to $X'$.
Let $f:Z\rightarrow X'$ be a closed subscheme, over $\bar k$, that is invariant under $F'_X$, meaning that there exists a $k'$-morphism $G:Z\rightarrow Z$ so that $f\circ G=F'_X\circ f$. Is it then true that $Z$ admits a descent to $k$?