  In this paper, we establish two main results which give conditions necessary
and sufficient for a $ 2 $-group metabelian such that $G/G'$ is of type $(2,
4)$ either metacyclic or not. If $G$ is the Galois group of
$\mathbf{k}_2^{(2)}/\mathbf{k}$ 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.
