Metacyclic $2$-extensions with cyclic kernel and the ultrasolvability questions

We give a necessary and sufficient conditions for $2$-local ultrasolvability of the metacyclic extensions. Then we derive the ultrasolvability for an arbibrary group extension, which has a local ultrasolvable associated subextension of the second type. Finally, using the above reductions, we establish the ultrasolvability results for a wide class of non-split $2$-extensions with cyclic kernel.