Automorphism groups of finite extensions of fields and the minimal ramification problem

We study the following question: given a global field F and finite group G, what is the minimal r such that there exists a finite extension K/F with Aut(K/F)≅G that is ramified over exactly r places of F? We conjecture that the answer is ≤1 for any global field F and finite group G. In the case when F is a number field we show that the answer is always ≤4[F:Q]. We show that assuming Schinzel's Hypothesis H the answer is always ≤1 if F is a number field. We show unconditionally that the answer is always ≤1 if F is a global function field. We also show that for a broader class of fields F than previously known, every finite group G can be realized as the automorphism group of a finite extension K/F (without restriction on the ramification). An important new tool used in this work is a recent result of the author and C. Tsang, which says that for any finite group G there exists a natural number n and a subgroup H⩽S_n of the symmetric group such that N_Sn(H)/H≅G.