TITLE: Profinite groups as absolute Galois groups
QUESTION [18 upvotes]: It is a well-known result that all profinite groups arise as the Galois group of some field extension.

What profinite groups are the absolute Galois group
  $\mathrm{Gal}(\overline{K}|K)$ of some extension $K$ over
  $\mathbb{Q}$?

The answer is simple enough in the finite case:

(Artin-Schreier) Only the trivial one and $C_2$.

This might tempt us to think that absolute Galois groups are not that diverse. But 0-dimensional anabelian geometry shows us differently:

(Neukirch-Uchida) There's as many different absolute Galois groups as non-isomorphic number fields.

The answer might still turn out to be boring, but I haven't seen this discussed anywhere. The closest is Szamuely in his book "Galois Groups and Fundamental Groups", where he seems delighted by the fact that the absolute Galois group of $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{q})$ are non-isomorphic for primes $p\neq q$.

REPLY [4 votes]: In order to study absolute Galois groups of fields one may also use another consequence of the aforementioned Rost-Voevodsky theorem: in fact, for any field $K$, the cohomology algebra $H^*(G_K,\mathbb{F}_p)$ is a quadratic algebra over the field $\mathbb{F}_p$ also for $p$ odd; and this is still true for the algebra $H^*(G_K(p),\mathbb{F}_p)$, where $G_K(p)$ denotes the maximal pro-$p$ quotient of $G_K$, if $K$ contains a primitive $p$-th root of 1 - i.e., $G_K(p)$ is the Galois group of the compositum of all Galois $p$-extensions $L/K$, and it is called the maximal pro-$p$ Galois group of $K$.
Note that the class of such Galois pro-$p$ groups includes all absolute Galois groups which are pro-$p$.
One reduces to pro-$p$ groups as in general they are easier to deal with than profinite groups.
Pro-$p$ groups whose $\mathbb{F}_p$-cohomology is a quadratic algebra - and thus they are «good candidates» for being realized as maximal pro-$p$ Galois groups - are studied by S. Chebolu, I. Efrat and J. Minàc in the paper Quotients of absolute Galois groups which determine the entire Galois cohomology (2012), and in my paper Bloch-Kato pro-$p$ groups and locally powerful groups (2014). Both papers are available also on the arXiv.
Also, recently I. Efrat, E. Matzri, J. Minàc and N.D. Tan proved that the $\mathbb{F}_p$-cohomology algebra of maximal pro-$p$ Galois groups of fields (containing a primitive $p$-th root of 1) satisfies another property, called the «vanishing of triple Massey products»: in the papers Triple Massey products and Galois theory and Triple Massey products vanish over all fields Minàc and Tan produce some examples of pro-$p$ groups which are not realizable as maximal pro-$p$ Galois groups (and thus as absolute Galois groups).
All the papers I mentioned are available in the arXiv: if you wish I may provide the links.