The Hilbert–Grunwald specialization property over number fields

Given a finite group G and a number field K, we investigate the following question: Does there exist a Galois extension E/K(t) with group G whose set of specializations yields solutions to all Grunwald problems for the group G, outside a finite set of primes? Following previous work, such a Galois extension would be said to have the “Hilbert–Grunwald property”. In this paper we reach a complete classification of groups G which admit an extension with the Hilbert–Grunwald property over fields such as K = ℚ. We thereby also complete the determination of the “local dimension” of finite groups over ℚ.