  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 $\sigma :G_{K_1,S_1}\to G_{K_2,S_2}$ an
isomorphism. Is $\sigma$ induced by a unique isomorphism between
$K_{1,S_1}/K_1$ and $K_{2,S_2}/K_2$?" Here the main assumption is about the
Dirichlet density of $S_i$.
