Negligible degree two cohomology of finite groups

For a finite group G, a G-module M and a field F, an element u ∈ H d ( G , M ) is negligible over F if for each field extension L / F and every group homomorphism Gal ( L sep / L ) → G , u belongs to the kernel of the induced homomorphism H d ( G , M ) → H d ( L , M ) . We determine the group of negligible elements in H 2 ( G , M ) for every abelian group M with trivial G-action.