Geyer in Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist, 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.

Geyer in Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist, 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 procyclic, see Satz 2.3 in the same paper.