${\mathbb{Z}}_2$-extension of real quadratic fields with $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ as $2$-class group at each layer

Let $K=\mathbb{Q}(\sqrt{d})$ be a real quadratic field with $d$ having three distinct prime factors. We show that the $2$-class group of each layer in the ${\mathbb{Z}}_2$-extension of $K$ is $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ under certain elementary assumptions on the prime factors of $d$. In particular, it validates Greenberg’s conjecture on the vanishing of the Iwasawa $\lambda$-invariant for a new family of infinitely many real quadratic fields.