Let k k be a global field and let K K be the maximal cyclotomic extension of k k . Let k ¯ {\overline {k}} be a separable closure of k k and G = G a l ( k ¯ / K ) G=\mathrm {Gal}({\overline {k}}/K) . For X X a smooth projective variety over k k , let X ¯ = X × k k ¯ {\overline {X}}=X\times _k{\overline {k}} . Ribet [Enseign. Math. 27 (1981), pp. 285–319] for i = 1 i=1 and Rössler-Szamuely [J. Reine Angew. Math. 752 (2019), pp. 211–227] for any odd i i showed that the fixed module of the étale cohomology group H i ( X ¯ , Q ℓ / Z ℓ ) H^i({\overline {X}},{{\mathbb {Q}}_{\ell }/{\mathbb {Z}}_{\ell }}) by G G is finite and zero for almost all ℓ ≠ c h a r ( k ) \ell \neq \mathrm {char}(k) , with a “large variation” assumption in positive characteristic. In this paper, building on their techniques, we consider the case of even degree cohomology, and prove positive and negative results in the number field and function field cases, respectively.