Let $K$ be a field, $K^{sep}$ its separable closure, and $G := Gal(K^{sep}/K)$ its absolute Galois group. What do we know about torsion in such groups? In particular, I am wondering if there is a restriction on the kind of subgroups, or if all finite subgroups can be embedded inside $G$.
If $K$ is a finite field then $G \cong \hat{\mathbb{Z}}$ is the profinite completion of $\mathbb{Z}$, in particular it is torsion-free. How about $K = \mathbb{Q}$? How about local fields like $K = \mathbb{Q}_p$?
The Artin-Schreier theorem says that for any field $K$ the only possibilities for torsion in $\operatorname{Aut}(\overline K/K)$ are in fact 2-torsion elements. See: https://kconrad.math.uconn.edu/blurbs/galoistheory/artinschreier.pdf
The absolute Galois group of $\mathbf Q$ does contain 2-torsion elements though, as we can choose an embedding $$\overline{\mathbf Q} \hookrightarrow \mathbf C$$ and act via complex conjugation to get an involution of $\overline{\mathbf Q}$ fixing $\mathbf Q$.