Subgroup Embeddings in a Finite Group: between Subnormality and $\mathrm{K}$-$\mathbb{P}$-Subnormality

Let $t$ be a chosen positive integer. A subgroup $H$ of a finite group $G$ is said to be $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal in $G$ if there exists a chain of subgroups
$$ H=H_{ 0} \leq H_{1} \leq \cdots \leq H_{m-1} \leq H_{m}=G $$
such that either $H_{i-1}$ is normal in $H_{i}$ or $|H_{i} : H_{i-1}|$ is some prime $p$ and $p-1$ is not divisible by $(t+1)$st powers of primes for any $i=1,\dots, m$. In this paper, properties of $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups are obtained. It is proved that all (supersolvable) groups with $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal Sylow subgroups form a hereditary saturated formation, and its local definition is found.