Geyer in [Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist][1], Satz 1.13 and the paragraph after that, gives a full characterization of which abelian profinite groups occur as absolute Galois groups: They are either $\mathbb{Z}/2\mathbb{Z}$ or $\prod_p\mathbb{Z}_p^{c(p)}$ for some cardinal numbers $c(p)$.

Side remark: For *algebraic* extensions of $\mathbb{Q}$, abelian absolute Galois groups are in fact cyclic, see Satz 2.3 in the same paper.


  [1]: https://doi.org/10.1016/0022-314X(69)90050-X