Greenberg's conjecture and Inverse Galois problem: metacyclic-nonmodular group of type 1 whose abelianization is $\mathbb{Z}/2 \mathbb{Z}\times\mathbb{Z}/2^m \mathbb{Z}$, $m\geq 2$

Abstract: For an integer $m\geq 2$, we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is $\mathbb{Z}/2 \mathbb{Z}\times\mathbb{Z}/2^m \mathbb{Z}$, as the Galois group of the maximal unramified $2$-extension (resp. pro-$2$-extension) of certain number fields of $2$-power degree (resp. cyclotomic $\mathbb Z_2$-extensions). Furthermore, we show up some new techniques for studying Greenberg's conjecture for some number fields. In particular, the reader can find results concerning the real quadratic fields $F=\mathbb{Q}(\sqrt{\eta q rs})$, the real biquadratic fields $K=\mathbb{Q}(\sqrt{\eta q},\sqrt{rs})$, with $\eta\in{1,2}$, and the Fr\"ohlich multiquadratic fields of the form $\mathbb F=\mathbb{Q}(\sqrt{q }, \sqrt {r}, \sqrt{s})$, where $q$, $r$ and $s$ are odd primes numbers.