Galois groups and cohomological functors

Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm{G}\mathrm{a}\mathrm{l}(F_{(3)}\mathrm{/}F)$ of $G_F$ which encodes the full mod-$q$ cohomology ring $H^{\ast }(G_F,\mathbb{Z}\mathrm{/}q)$ and is minimal with respect to this property. We prove some fundamental structure theorems related to these quotients. In particular, it is shown that when $q=p$ is an odd prime, $F_{(3)}$ is the compositum of all Galois extensions $E$ of $F$ such that $\mathrm{G}\mathrm{a}\mathrm{l}(E\mathrm{/}F)$ is isomorphic to $\{1\}$, $\mathbb{Z}\mathrm{/}p$ or to the nonabelian group $H_{p^3}$ of order $p^3$ and exponent $p$.