On the structure and stability of ranks of $2$-class groups in cyclotomic ${\mathbb{Z}}_2$-extensions of certain real quadratic fields

For a real quadratic field $K=\mathbb{Q}(\sqrt{d})$ with discriminant $D_K$ having four distinct prime factors, we study the structure of the $2$-class group $A(K_1)$ of the first layer $K_1=\mathbb{Q}(\sqrt{2},\sqrt{d})$ of the cyclotomic ${\mathbb{Z}}_2$-extension of $K$. With some suitably convenient assumptions on the rank and the order of $A(K_1)$, we characterize $K$ for which the $2$-class group $A(K)$ is isomorphic to $\mathbb{Z}\mathrm{/}2\mathbb{Z}\oplus \mathbb{Z}\mathrm{/}2\mathbb{Z}$. We infer that the $2$-ranks of the class groups in each layer stabilizes by virtue of a result of Fukuda. This also provides an alternate way to establish that the Iwasawa $\mu$-invariant of $K$ vanishes. In some cases, we also provide sufficient conditions on the constituent prime factors of $D_K$ that imply $A(K)\simeq \mathbb{Z}\mathrm{/}2\mathbb{Z}\oplus \mathbb{Z}\mathrm{/}2\mathbb{Z}$, $A(K_1)\simeq \mathbb{Z}\mathrm{/}2\mathbb{Z}\oplus \mathbb{Z}\mathrm{/}4\mathbb{Z}$ and $A(K^{\mathrm{\prime }})\simeq \mathbb{Z}\mathrm{/}2\mathbb{Z}\oplus \mathbb{Z}\mathrm{/}2\mathbb{Z}\oplus \mathbb{Z}\mathrm{/}2\mathbb{Z}$, where $K^{\mathrm{\prime }}=\mathbb{Q}(\sqrt{2d})$. This extends some results obtained by Mizusawa.