  Let $d$ be a positive square-free integer. In this paper we shall investigate
the structure of the $2$-class group of the cyclotomic $\mathbb{Z}_2$-extension
of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$.
Furthermore, we deduce the structure of the $2$-class group of cyclotomic
$\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{-d})$.
