Composita of symmetric extensions of Q

Let K be a Hilbertian presented field with elimination theory of characteristic
≠ 2, let Ksymm be the compositum of all symmetric extensions of K, and let Ksymm,ins be the maximal purely inseparable extension of Ksymm. Then, Th(Ksymm,ins) is a primitive recursive theory. Moreover, the set of finite groups that can be realized as Galois groups over K in Ksymm as well as the set of finite groups that occur as Galois groups over Ksymm are primitive recursive subsets of the set of all finite groups. Finally, if K is countable, then Gal(Ksymm/K) ∼= Gal(Qsymm/Q).