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We study the capitulation of $2$-ideal classes of an infinite family ofimaginary bicyclic biquadratic number fields consisting of fields $\mathbf{k}=\mathbb{Q}(\sqrt{p_1p_2q}, i)$, where $i=\sqrt{-1}$ and $p_1\equivp_2\equiv-q\equiv1 \pmod 4$ are different primes. For each of the threequadratic extensions $\mathbf{K}/\mathbf{k}$ inside the absolute genus field$\mathbf{k}^{(*)}$ of $\mathbf{k}$, we compute the capitulation kernel of$\mathbf{K}/\mathbf{k}$. Then we deduce that each strongly ambiguous class of$\mathbf{k}/\mathbb{Q}(i)$ capitulates already in $\mathbf{k}^{(*)}$, which issmaller than the relative genus field $(\mathbf{k}/\mathbb{Q}(i))^*$.

Capitulation in the absolutely abelian extensions of some fields $\mathbb{Q}(\sqrt{p_1p_2q}, \sqrt{-1})$