Let $K$ be a number field. I am wondering if the following exact sequence $$1 \longrightarrow[\widehat{\mathcal O}\_K^\times] \longrightarrow Gal(K^{ab}/K) \overset{\pi}{\longrightarrow} Cl\_K \longrightarrow 1$$ splits, i.e. if there is a homomorphism $s:Cl\_K \to Gal(K^{ab}/K)$ such that $s \circ \pi = id$. Here $[\widehat{\mathcal O}\_K^\times]$ stands for the image of $\widehat{\mathcal O}\_K^\times$ in $Gal(K^{ab}/K)$ under Artin's reciprocity morphism and $Cl\_K$ denotes the ideal class group of $K$. 


Equivalently we can rewrite the above sequence idelically as


$$1 \to \widehat{\mathcal O}\_K^\times / \overline{\mathcal O \_{K,+} ^\times} \to \widehat{\mathcal O}\_K^\natural / \overline{\mathcal O \_{K,+}^\natural} \to \widehat{\mathcal O}\_K^\natural / (\widehat{\mathcal O}\_K^\times \cdot \mathcal O \_{K,+}^\natural) \to 1,$$
where $\mathcal O \_K ^\natural = \mathcal O \_K - \{ 0 \} $, $\widehat{\mathcal O}\_K^\natural = \mathbb A \_{K,f}^\times \cap \widehat{\mathcal O}\_K$. Further, subscript $+$ denotes totally positive elements and over lined objects are meant to be closures in the idele topology. (I don't know if this is helpful.) 



On the structure of the maximal abelian Galois group of a number field
Let $K = {\mathbb Q}(\sqrt{-5})$ and let $P = (3+2\sqrt{-5})$ denote a prime ideal of norm $29$ in the ring of integers of $K$. There does not exist a quadraic extension of $K$ ramified exactly at $P$, but there is one over the Hilbert class field $K(i)$ of $K$. This means that the ray class group modulo $P$ does not split into the product of the ideal class group, which has order $2$, and another group. In particular, the maximal abelian extension unramified outside of $P$ does not split into a compositum of an unramified extension and a purely ramified extension, and neither does $K^{\rm{ab}}$.


