It is well-known that any finite group is the Galois group of a Galois extension. This follows from Cayley's theorem (as can be seen in this answer). This (linked) answer led me to the following question:
What about infinite groups?
Infinite groups appear as Galois groups on infinite extensions, as when one defines the absolute Galois group of a field $F$. There is a natural topology on the Galois group, called the Krull topology, which turns this group into a profinite group (i.e. a compact, totally disconnected, Hausdorff topological group). It can be proven that any profinite group is the Galois group of an extension (see this short paper by Waterhouse). Therefore, the above question is equivalent to the following:
Can any group be given the structure of a profinite group?
I would like to get more information about this question. In particular, if the answer is no, are there definite restrictions (e.g. cardinality)?