Group extensions with a non-commutative kernel

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.

When the kernel $K$ is non-abelian, as you rightly say there is no conjugation action of $H$ on $K$, only an outer action, ie a homomorphism $\psi\colon\thinspace H\to \mathrm{Out}(K) := \mathrm{Aut}(K)/\mathrm{Inn}(K)$. Then extensions which induce this outer action are classified up to equivalence by $H^2(H;ZK)$, where $ZK$ denotes the center of $K$.
This should also be linked to Dedecker's notions of cohomology with coefficients in a crossed module, where the usual theory is for the crossed module $K \to Aut(K)$, and also to a paper of Turing linking with identities among relations. See our paper on the Schreier theory, Proceedings Royal Irish Academy 96A (1996) 213-227.. The link to crossed complexes is also stressed in the book Nonabelian Algberaic Topology, Chapter 12; this link allows the use of model category type techniques, such as fibrations of crossed complexes, and tensor products of crossed complexes for calculational purposes.