Structure of the Galois group of the maximal unramified pro-$2$-extension of some $\mathbb{Z}_2$-extensions

For a number field $ k$, we consider the Galois group $G=\operatorname{Gal}(\mathcal{L}( k_{\infty})/ k_{\infty})$ of the maximal unramified pro-$2$-extension of the cyclotomic $\mathbb{Z}_2$-extension $ k_{\infty}$ of $ k$. In terms of transfer, we establish a necessary and sufficient condition for a $2$-group to be abelian or metacyclic non-abelian whenever its abelianization is of type $(2^n, 2^m)$, with $n\geq2$ and $m\geq2$. Then we apply this result to construct an infinite family of real quadratic fields for which $G$ is an abelian pro-$2$-group of rank $2$.