Sur l'application de Quillen pour la cohomologie modulo 2 de certains groupes finis

Let $G$ be a finite group. In a famous article, Quillen describes an $\mathrm{F}$-isomorphism between commutative $\mathbb{N}$-graded ${\mathbb{F}}_2$-algebras $${\mathrm{q}}_G:{\mathrm{H}}^{\ast }(G;{\mathbb{F}}_2)\to \mathrm{L}(G),$$ with $\mathrm{L}(G)$ defined in terms of the elementary abelian $2$-groups contained in $G$. We show that ${\mathrm{q}}_G$ is in fact an actual isomorphism for three families of finite groups: symmetric groups, alternating groups and finite Coxeter groups.