Let $X$ be a quasi-projective variety over a finite field $\mathbb F_q$ where $q$ is a power of a prime $p$. Let $\ell$ be a prime number different from $p$. We have the relative Frobenius morphism $$\mathcal F: X \to X^{(p)},$$ as defined for instance in the stack project here.
How do the cohomology groups $H^i(X^{(p)} \times \overline{\mathbb F_p}, \mathbb Q_{\ell})$ compare with $H^i(X\times \overline{\mathbb F_p}, \mathbb Q_{\ell})$?