Specializations of regular Galois extensions with prescribed local behaviour

The Grunwald Problem asks whether, for given local extensions (all with Galois group embedding into some fixed group G) at finitely many primes of a number field K, there always exists a G-extension of K which approximates all these local extensions. We investigate to what extent the specializations of some given regular Galois extensions can provide positive answers to this question. We also give application to other problems in Galois theory, such as the question of existence of parametric G-extensions and the problem of K-admissibility of finite groups.