Is every group a Galois group?

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)?

No. Profinite groups are residually finite (in fact a group is residually finite if and only if it embeds into its profinite completion) and many groups are not residually finite. If you don't have any particular restriction on the number of generators, $\mathbb{Q}$ is a simple example.

There are other restrictions. A compact Hausdorff group has a Haar measure of total measure $1$ and no countable group can be equipped with such a measure since the measure of any singleton cannot be $0$ and cannot be positive. This rules out $\mathbb{Q}$ but it also rules out, for example, $\mathbb{Z}$.

This cannot be true since a profinite group cannot be countable (see this) so therefore we cannot make $\mathbb{Z}$ into a profinite group.