Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$.

Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive dimension?

The reason why I ask, is that when $X$ is a complex algebraic variety then the vector space of rational Hodge classes is always nonzero because it contains a power of the class of a Kähler form.

Can one argue in a similar way using the class of a hyperplane section in $H^2(X,\mathbf{Q}_{\ell}(1))$?

Can one deduce that the group of con dimension $k$-cycles modulo homological equivalence on $X$ is always nonzero?