On unramified Galois 2-groups over \(\mathbb{Z}_2\)-extensions of some imaginary biquadratic number fields

For an imaginary biquadratic number field \(K = \mathbb Q(\sqrt{-q},\sqrt d)\), where \(q>3\) is a prime congruent to \(3 \pmod 8\), and \(d\) is an odd square-free integer which is not equal to q, let \(K_\infty\) be the cyclotomic \(\mathbb Z_2\)-extension of \(K\). For any integer \(n \geq 0\), we denote by \(K_n\) the nth layer of \(K_\infty/K\). We investigate the rank of the 2-class group of \(K_n\), then we draw the list of all number fields K such that the Galois group of the maximal unramified pro-2-extension over their cyclotomic \(\mathbb Z_2\)-extension is metacyclic pro-2 group.