Groups with many pronormal and transitively normal subgroups

A subgroup $H$ of a group $G$ is said to be transitively normal in $G$, if $H$ is normal in every subgroup $K\geq H$ such that $H$ is subnormal in $K$. The study of radical groups, whose not finitely generated subgroups are transitively normal, has been started by L. A. Kurdachenko, N. N. Semko (Jr.), I. Ya. Subbotin. In this paper the study of such groups is continued.