Capitulation in the absolutely abelian extensions of some number fields II

We study the capitulation of $2$-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields $k =Q(\sqrt{pq_1q_2}, i)$,where$i=\sqrt{-1}$and$q_1\equiv q_2\equiv-p\equiv-1 \pmod 4$ are different primes. For each of the three quadratic extensions $K\mathrm{/}k$ inside the absolute genus field $k^{(\ast )}$ of $k$, we compute the capitulation kernel of $K\mathrm{/}k$. Then we deduce that each strongly ambiguous class of $k\mathrm{/}Q(i)$ capitulates already in $k^{(\ast )}$.