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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 itsGalois group. In this paper, we answer the following question under someassumptions: "For $i=1,2$, let $K_i$ be a number field, $S_i$ a (sufficientlylarge) set of primes of $K_i$ and $\sigma :G_{K_1,S_1}\to G_{K_2,S_2}$ anisomorphism. 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 theDirichlet density of $S_i$.

Isomorphisms of Galois groups of number fields with restricted ramification