I am looking for examples $(X,\eta)$ where the integral hard Lefschetz is an isomorphism:
$X/k$ is a smooth projective variety of dimension $d$ over an algebraic closure of a finite field and $\eta \in H^2(X,\mathbf{Z}_\ell(1))$ the class of an ample hypersurface section. Consider all cohomology groups modulo torsion. By the (rational) hard Lefschetz, $(\cup\eta)^{d-1}: H^1(X,F) \to H^{2d-1}(X,F(d-1))$, with $F$ an $\ell$-adic sheaf, is injective with finite cokernel. For $F$ take $\mathbf{Z}_\ell$ or $T_\ell\mathscr{A}$ with $\mathscr{A}/X$ an Abelian scheme.
For example, I this holds if $X = \mathbf{P}^d_k$ and $F = \mathbf{Z}_\ell$. (OK, this is clear since $H^1(\mathbf{P}^d_k,\mathbf{Z}_\ell) = 0$.)
If it holds for $F = \mathbf{Z}_\ell$, it holds for $F = T_\ell\mathscr{A}$ if $\mathscr{A} = A \times_k X$ is constant, so I am interested in cases where $\mathscr{A}/X$ is not constant.