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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 $A(K)$ isisomorphic to $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. We inferthat the $2$-ranks of the class groups in each layer stabilizes by virtue of aresult of Fukuda. This also provides an alternate way to establish that theIwasawa $\mu$-invariant of $K$ vanishes. In some cases, we also providesufficient conditions on the constituent prime factors of $D_{K}$ that imply$A(K) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, $A(K_{1})\simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z}$ and $A(K^{\prime})\simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus\mathbb{Z}/2\mathbb{Z}$, where $K^{\prime} = \mathbb{Q}(\sqrt{2d})$. Thisextends some results obtained by Mizusawa.

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