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

A survey of solvability conditions for the embedding problem of number fields, in which the kernel is a non-Abelian group of order $p^4$, is completed. As a kernel, the two $2$-groups with two generators $a,b$ and with the following relations are considered: $a^8=1$, $b^2=1$, $[a,b]=a^{-2}$ in the first group, and $a^8=1$, $b^2=a^4$, $[a,b]=a^2$ in the second. Bibliography: 7 titles.