On the embedding problem with noncommutative kernel of order $p^4$. VI

In the case of number fields the embedding problem of a $p$-extension with non-Abelian kernel of order $p^4$ is studied. The two kernels of order $3^4$ with generators $\alpha,\gamma$ and relations $\alpha^9=1$, $[\alpha,\gamma]^3=1$, $[\alpha,\alpha,\gamma]=1$, $[\alpha,\gamma,\gamma]=\alpha^3$, $\gamma^3=1$ or $\gamma^3=\alpha^3$, and the kernel of order $2^4$ with generators $\alpha,\beta,\gamma$ and relations $\alpha^4=1$, $\beta^2=\gamma^2$, $[\alpha,\beta]=1$, $[\alpha,\gamma]=1$, $[\beta,\gamma]=\alpha^2$ are considered. For kernels of odd order the embedding problem is always solvable. For the kernel of order 16 the solvability conditions are reduced to those for the associated problems at the Archimedean points, and to the compatibility condition. Bibliography: 9 titles.