On the Existence of the Maximal Unramified Pro-$2$-Extension over the Cyclotomic ${\mathbb{Z}}_2$-Extension with Prescribed Metacyclic Galois Group

For an integer $m\ge 2$, we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is $\mathbb{Z}\mathrm{/}2\mathbb{Z}\times \mathbb{Z}\mathrm{/}{2}^m\mathbb{Z}$, as the Galois group of the maximal unramified $2$-extension (resp. pro-$2$-extension) over certain number fields of $2$-power degree (resp. cyclotomic ${\mathbb{Z}}_2$-extensions). Furthermore, we present some new techniques for studying Greenberg’s conjecture for some number fields. In particular, the reader can find results concerning the real quadratic fields $F=\mathbb{Q}(\sqrt{\eta qrs})$, the real biquadratic fields $K=\mathbb{Q}(\sqrt{\eta q},\sqrt{rs})$, with $\eta \in \{1,2\}$, and the Fröhlich multiquadratic fields of the form $\mathbb{F}=\mathbb{Q}(\sqrt{q},\sqrt{r},\sqrt{s})$, where $q$, $r$ and $s$ are odd prime numbers.