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Let $K$ be a number field, $G$ finite group and $L/K$ a $G$-Galois extension. Denote $\mathcal{O}_K$ and $\mathcal{O}_L$ the respective rings of integers.

By the following result of Auslander and Buchsbaum, we are able to recover a $G$-Galois extension of the ring of integers of $K$ with restricted ramification $\mathcal{O}_{K,S}$, for $S$ being a set of finite places, from every $G$-Galois extension of $K$ .

Let $L/K$ be as above, let S be a set of finite places of $K$ and $S'$ be the set of places in $L$ dividing some $\mathfrak{p}\in S$. Then $\mathcal{O}_{L,S'}/\mathcal{O}_{K,S}$ is a $G$-Galois extension of rings iff $L/K$ is unramified outside $S$.

My question is the following:

Is it true that every $G$-Galois extension of $\mathcal{O}_{K,S}$ corresponds uniquely to a $G$-Galois extension of $K$ which unramified outside $S$?

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