  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 $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. 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
$A(K) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, $A(K_{1})
\simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z}$ and $A(K^{\prime})
\simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus
\mathbb{Z}/2\mathbb{Z}$, where $K^{\prime} = \mathbb{Q}(\sqrt{2d})$. This
extends some results obtained by Mizusawa.
