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
$$H = H_0 \subset H_1 \subset \ldots \subset H_n = G$$
such that $|H_i:H_{i-1}|$ is a prime for every $i = 1, 2,\ldots, n$. 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.