On the embedding problem with non-Abelian kernel of order $p^4$. IV

In the present paper, the embedding problem is considered for number fields with $p$-groups whose kernel is either of two groups with two generators $\alpha$ and $\beta$ and with the following relations:
(1) $\alpha^p=1$, $\beta^p=1$, $[\alpha,\beta,\beta]=1$, $[\alpha,\beta,\alpha,\alpha]=1$ or
(2) $\alpha^p=[\alpha,\beta,\alpha]$, $\beta^p=1$, $[\alpha,\beta,\beta]=1$.
It is shown that for the solvability of the original embedding problem it is necessary and sufficient to have the solvability of the associated Abelian and local problems for all completions of the base fields. Bibliography: 7 titles.