Isomorphisms of Galois groups of number fields with restricted ramification

Let K be a number field and S a set of primes of K. We write K-S/K for the maximal extension of K unramified outside S and G (K,S) for its Galois group. In this paper, we answer the following question under some assumptions: For i = 1, 2, let K-i be a number field, S-i a (sufficiently large) set of primes of K-i and sigma: G(K1),S-1 -> G(K2),S-2 an isomorphism. Is ?? induced by a unique isomorphism between K-1S1/K-1 and K-2,K-S2,/K-2? Here, the main assumption is about the Dirichlet density of Si.