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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.

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Group extensions are classified by outer actions of the cokernel on the kernel together with third and second cohomology groups. All this is in an old paper by MacLane –  Fernando Muro May 5 '13 at 15:40
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2 Answers

In response to your first question: yes, the theory of extensions with non-abelian kernel has been developed, by Eilenberg, Mac Lane and others. It is summarized in Section IV.6 of Ken Brown's "Cohomology of groups" book, where you will find references to the original literature.

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 $K$.

See also the reference linked to in Peter Michor's answer to my recent question, which may contain more relevant information.

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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.

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