  Let $p_1 \equiv p_2 \equiv5\pmod8$ be different primes. Put $i=\sqrt{-1}$ and
$d=2p_1p_2$, then the bicyclic biquadratic field $k=Q(\sqrt{d}, \sqrt{-1})$ has
an elementary abelian 2-class group of rank $3$. In this paper we determine the
nilpotency class, the coclass, the generators and the structure of the
non-abelian Galois group $\mathrm{Gal}(k_2^{(2)}/k)$ of the second Hilbert
2-class field $k_2^{(2)}$ of $k$. We study the capitulation problem of the
2-classes of $k$ in its seven unramified quadratic extensions $K_i$ and in its
seven unramified bicyclic biquadratic extensions $L_i$.
