On ultrasolvability of $p$-extensions of an abelian group by a cyclic kernel

We solve a problem in the embedding theory by A. V. Yakovlev for $p$-extensions of odd order with cyclic normal subgroup and abelian quotient-group: for such nonsplit extensions there exists a realization for the quotient-group as Galois group over number fields such as corresponding embedding problem is ultrasolvable (i.e. this embedding problem is solvable and has only fields as solutions). Also we give a solution for embedding problems of $p$-extensions of odd order with kernel of order $p$ and with a quotient-group which is represented by direct product of its proper subgroups – this is a generalization for $p>2$ an analogous result for $p=2$ by A. Ledet.