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.
Categories: math.NT