The main object of the paper under review is the fundamental group \(G_S=\pi_1(\operatorname{Spec} O_{K,S})\) where \(K\) is a number field and \(S\supseteq S_{\infty}\) is a finite set of primes of \(K\). The author is interested in reconstructing the position of the decomposition groups \(D_{\mathfrak p}\), \(\mathfrak p\in S\setminus S_{\infty}\), inside \(G_S\). In contrast to a classical theorem by J. Neukirch [Invent. Math. 6, 296–314 (1969; Zbl 0192.40102)] for \(G_K=\operatorname{Gal}(K)\), saying that this group fully encodes such information for the decomposition groups of primes of \(K\), the author shows that some arithmetical information is to be added. He exhibits several equivalent ways for adding such data: the cyclotomic character on \(G_S\); the \(S\)-class number of all finite subfields \(K_S/L/K\); the number of primes of \(L\) lying over \(S\) for all these \(L\).