Isomorphisms of Galois groups of number fields with restricted ramification

Let $K$ be a number field and $S$ set of primes $K$. We write $K_S/K$ for the maximal extension unramified outside $G_{K,S}$ its Galois group. In this paper, we answer following question under some assumptions: "For $i=1,2$, let $K_i$ field, $S_i$ (sufficiently large) $\sigma :G_{K_1,S_1}\to G_{K_2,S_2}$ an isomorphism. Is $\sigma$ induced by unique isomorphism between $K_{1,S_1}/K_1$ $K_{2,S_2}/K_2$?" Here main assumption is about Dirichlet density $S_i$.