An analog of Quillen-Venkov for eventually negligible classes and radicality for finite irreducible Coxeter groups

Let $G$ be a finite group, $M$ a $G$-module, and $F$ a field. A continuous group homomorphism $i : Γ_L=\text{Gal}(L_{\text{sep}}/L) \rightarrow G$ yields a homomorphism $i^* : H^d(G,M) \rightarrow H^d(L,M)$ for every $d\in \mathbb{N}$. An element $u \in H^d(G,M)$ is negligible over $F$ if $u\in\ker(