Non-algebraic geometrically trivial cohomology classes over finite fields

We give the first examples of smooth projective varieties $X$ over a finite field $\mathbb{F}$ admitting a non-algebraic torsion $\ell$-adic cohomology class of degree $4$ which vanishes over $\overline{\mathbb{F}}$. We use them to show that two versions of the integral Tate conjecture over $\mathbb{F}$ are not equivalent to one another and that a fundamental exact sequence of Colliot-Th\'el\`ene and Kahn does not necessarily split. Some of our examples have dimension $4$, and are the first known examples of fourfolds with non-vanishing $H^{3}_{\text{nr}}(X,\mathbb{Q}_{2}/\mathbb{Z}_{2}(2))$.