  Let $d$ be a square-free integer, $\mathbf{k}=\mathbb{Q}(\sqrt d,\,i)$ and
$i=\sqrt{-1}$. Let $\mathbf{k}_1^{(2)}$ be the Hilbert $2$-class field of
$\mathbf{k}$, $\mathbf{k}_2^{(2)}$ be the Hilbert $2$-class field of
$\mathbf{k}_1^{(2)}$ and $G=\mathrm{Gal}(\mathbf{k}_2^{(2)}/\mathbf{k})$ be the
Galois group of $\mathbf{k}_2^{(2)}/\mathbf{k}$. Our goal is to give necessary
and sufficient conditions to have $G$ metacyclic in the case where $d=pq$, with
$p$ and $q$ are primes such that $p\equiv 1\pmod 8$ and $q\equiv 5\pmod 8$ or
$p\equiv 1\pmod 8$ and $q\equiv 3\pmod 4$.
