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

For an imaginary biquadratic number field \(K = \mathbb Q(\sqrt{-q},\sqrt d)\) $K=\mathbb{Q}(\sqrt{-q},\sqrt{d})$ , where \(q>3\) $q>3$ is a prime congruent to \(3 \pmod 8\) $3(mod8)$ , and \(d\) $d$ is an odd square-free integer which is not equal to q, let \(K_\infty\) $K_{\infty }$ be the cyclotomic \(\mathbb Z_2\) ${\mathbb{Z}}_2$ -extension of \(K\) $K$ . For any integer \(n \geq 0\) $n\ge 0$ , we denote by \(K_n\) $K_n$ the nth layer of \(K_\infty/K\) $K_{\infty }/K$ . We investigate the rank of the 2-class group of \(K_n\) $K_n$ , 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 \(\mathbb Z_2\) ${\mathbb{Z}}_2$ -extension is metacyclic pro-2 group.