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 $K=\mathbb{Q}$. We thereby also complete the determination of the ``local dimension" of finite groups over $\mathbb{Q}$.