Ultrasoluble coverings of some nilpotent groups by a cyclic group over number fields and related questions

-Let $F$ be a finite nilpotent group of odd order. For every finite cyclic subgroup $A$ of odd order we find necessary and sufficient conditions for a class $h\in H^2(F,A)$ to determine an ultrasoluble extension (under the additional assumption of minimality of all $p$-Sylow subextensions to the extension with class $h$ for all non-Abelian $p$-Sylow subgroups $F_p$ of $F$), that is, for the existence of a Galois extension of number fields $K/k$ with group $F$ such that the corresponding embedding problem is ultrasoluble (it has solutions and all such solutions are fields). We also establish a number of related results.