Capitulation des $2$-classes d'id\'eaux de type $(2,4)$

In this paper, we establish two main results which give conditions necessary and sufficient for a $2$-group metabelian such that $G\mathrm{/}{G}^{\mathrm{\prime }}$ is of type $(2, 4)$eithermetacyclicornot\mathrm{.}If$G$istheGaloisgroupof$$\mathbf{k}$_2^{(2)}/$\mathbf{k}$$where$$\mathbf{k}$_2^{(2)}$ is the the Second Hilbert $2$-class field of a number field $\mathbf{k}$, we will get results on the problem of capitulation.