Profinite version of the Beckmann-Black problem

The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point t0∈P1(K) of some finite regular Galois extension F/K(T) with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the l-universal Frattini cover of an arbitrary finite group.