On the strongly ambiguous classes of $k\mathrm{/}Q(\sqrt{-1})$ where $k=Q(\sqrt{2p_1{p}_2},\sqrt{-1})$

We construct an infinite family of imaginary bicyclic biquadratic number fields $k$ with the 2-ranks of their 2-class groups are $\ge 3$, whose strongly ambiguous classes of $k\mathrm{/}Q(i)$ capitulate in the absolute genus field $k^{(\ast )}$, which is strictly included in the relative genus field $(k\mathrm{/}Q(i){)}^{\ast }$ and we study the capitulation of the $2$-ideal classes of $k$ in its quadratic extensions included in $k^{(\ast )}$.