Y. Mizusawa [J. Théor. Nombres Bordx. 22, No. 1, 115–138 (2010; Zbl 1221.11215)] computed a pro-\(2\) presentation of the Galois group of the maximal unramified pro-\(2\)-extension of the cyclotomic \(\mathbb Z_2\)-extension of certain complex quadratic number fields. In this article it is shown that the same methods allow to discuss some maximal tamely ramified pro-\(2\)-extensions. The author’s main result is the following: let \(p \equiv \pm 3 \bmod 8\) and \(q \equiv -p \bmod 8\) be prime numbers and put \(S = \{q\}\). Let \(k = \mathbb Q(\sqrt{-p}\,)\), let \(k_\infty\) be the cyclotomic \(\mathbb Z_2\)-extension of \(k\), and \(L_S^\infty\) the maximal pro-\(2\)-extension of \(k_\infty\) unramified outside of \(S\). Then the Galois group \(G\) of \(L_S^\infty/k_\infty\) has rank \(2\), its abelianization is isomorphic to \(\mathbb Z_2 \oplus \mathbb Z/2\mathbb Z\) as a \(\mathbb Z_2\)-module, and \(G\) has the presentation \(G = \langle a, b| [a,b]a^2 \rangle\), where \([a,b] = a^{-1}b^{-1}ab\) is the commutator of \(a\) and \(b\).