Let

$$ 1\to K\to G\to H\to 1 $$ be an extension of groups. When $K$ is commutative, $H$ acts on $K$ by conjugation; and given groups $K$ and $H,$ with $K$ commutative and $H$ acting on $K,$ such extensions are classified by group cohomology $H^2.$ For instance, if $H$ is profinite and $K$ is a (commutative) discrete $H$-module, then any extension splits potentially, i.e. $G\to H$ has a section over an open subgroup of $H,$ as the continuous group cohomology equals the direct limit of cohomologies of finite groups.

Is there any theory when $K$ is not commutative? In this case, an extension does not induce a "conjugation" action of $H$ on $K.$ For instance, when $K=G(k^s)$ for some algebraic group $G$ over a field $k$ and $H=Gal(k^s/k),$ it seems that one may still ask for "affine extensions with kernel $G$" when $G$ is not a torus, just no $H^2$-interpretation.

Let me ask a real question. Let $K_N$ be the maximal extension of $\mathbb Q$ unramified outside $N$ (for an integer $N>1$), and let $G=Gal(K_N/\mathbb Q).$ Consider its abelianization $H,$ which is $\prod_{p|N}\mathbb Z_p^*$ by CFT. Does the projection $G\to H$ have a section over some open subgroup of $H?$

Thank you.