Finite Groups with $$\mathbb{P}$$-Subnormal Schmidt Subgroups

A subgroup $$H$$ of a group $$G$$ is called $$\mathbb{P}$$ -subnormal in $$G$$ whenever either $$H=G$$ or there is a chain of subgroups $$H=H_{0}\subset H_{1}\subset\mathinner{\ldotp\ldotp\ldotp}\subset H_{n}=G$$ such that $$|H_{i}:H_{i-1}|$$ is a prime for every $$i=1,2,\mathinner{\ldotp\ldotp\ldotp},n$$ . We study the structure of a finite group $$G$$ all of whose Schmidt subgroups are $$\mathbb{P}$$ -subnormal. The obtained results complement the answer to Problem 18.30 in the Kourovka Notebook..