Abelian extensions of algebraic extensions of $\Bbb Q$ Let $F/\Bbb Q$ be an algebraic extesnion (may not be finite), do we have class field theory for abelian extensions of $F$ ? In other words, can we describe $G^{ab}_F$?
If $F$ contains all roots of unity, then by Kummer's theory we know the pro-p quotient $G_F^{ab}(p) \cong \Bbb Z_p^{dim_{\Bbb F_p}F^{\times}/(F^{\times})^p}$, see remarks on maximal abelian extensions 2.9. in this article. But this isomorphism is not useful in practice and does not have many good properties compared to the Artin homomorphism.
 A: If $[F:\mathbf Q]$ is finite, i.e. $F$ is a number field, the description of the finite abelian extensions of $F$ and their arithmetic is in theory entirely assured by CFT, in ideal or in idelic terms, see a summary e.g. here https://math.stackexchange.com/a/2127486/300700. To go to infinite abelian extensions, in principle one can take inductive limits of fields (resp. projective limits of Galois groups). To avoid stumbling on non-noetherian modules, one usually "restricts ramification", i.e. one considers abelian extensions which are unramified outside a finite set $S$ of primes of $F$. This amounts to study, not the whole $G_F^{ab}$ (= Galois group of the maximal abelian extension of $F$), but the quotient $G_{F,S}^{ab}$ (= Galois group of the maximal abelian extension of F which is unramified ouside $S$), or even its maximal pro-$p$-quotient $X_S(F)$, with $S$ containing all the $p$-places of $F$. Then $X_S(F)$ is a noetherian $\mathbf Z_p$-module, but even then not everything is known. The precise $\mathbf Z_p$-rank of $X_S(F)$ is given by the famous Leopoldt conjecture (solved if $F$ is abelian over $\mathbf Q$), and the $\mathbf Z_p$-torsion submodule is - roughly speaking - related to the $p$-adic L-functions attached to $F$ when $F$ is totally real (this is the so called Main Conjecture of Iwasawa theory, now the theorem of Mazur-Wiles). For a glimpse, see e.g. https://math.stackexchange.com/a/1898156/300700.
Now what seems to interest you most is about the sequence of $F < F^{ab} < (F^{ab})^{ab} <...$ In terms of Galois groups, if $G = G_F$ is the absolute Galois group of the number field $F$, you woud like to know about the successive quotients of $G$ modulo the successive subgroups of its descending derived series. Again to avoid problems of non-noetherianity, one usually take the $p$-adic point of view and rather deals with the $q$-descending central series of $G$, where $q$ is a power of a prime $p$ : $G^{(1)}=G, G^{(i+1)}=G^q[G^{(i)}, G]$ for $i=1, 2,...$, and $G^{[i]}=G/G^{(i)}$. The surprising result obtained by Jan Minac and his collaborators (around 2012) is that for any field of characteristic $\neq p$ and containing a primitive $q$-th root of unity (the so called kummerian situation), $G^{[3]}$ is uniquely determined by $H^1 (G, \mathbf Z/q)$ and by the cup-product on $H^1(G,\mathbf Z/q)\otimes H^1(G,\mathbf Z/q)$. For $q=p$, this yields an explicit Galois description  of the field $F_3$ fixed by $G^{(3)}$ inside a separable closure of $F$. 
Of course when $F$ is a number field, more precise results can be obtained by playing with arithmetical properties. To come back to the example given in your post, take $F=\mathbf Q , K=\mathbf Q^{ab}, H=G_F^{ab}, p=2$. Then an $\mathbf F_2$-basis of ${(K^*/{K^*}^2)}^H$ is formed of the classes mod ${K^*}^2$ of $\sqrt l\cup {sin[a_{q,r}]}$, where $l$ runs through all primes and $(r,q)$ runs through all pairs of primes $r<q$ ; here $sin [a]=2 sin\pi a$ if $0<a<1, a\in \mathbf Q$ , $1$ if $a=0$, and the $a_{q,r}$ are complicated sums of rationals... which I don't know how to type ! This was a theorem of G. W. Anderson (2002), called "Kronecker-Weber + $\epsilon$" ./.
