  Let $d$ be an odd square-free integer, $m\geq 3$ any integer and $L_{m,
d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all
the fields $L_{m, d}$ having an odd class number. Furthermore, using the
cyclotomic $\mathbb{Z}_2$-extensions of some number fields, we compute the rank
of the $2$-class group of $L_{m, d}$ whenever the prime divisors of $d$ are
congruent to $3$ or $5\pmod 8$.
