On $\mathrm{K}$-${\mathbb{P}}_t$-subnormal subgroups of finite groups and related formations

Let $t$ be a fixed natural number. A subgroup $H$ of a group $G$ will be called $\mathrm{K}$-${\mathbb{P}}_t$-subnormal in $G$ if there exists a chain of subgroups $H=H_0\le H_1\le \cdots \le H_{m-1}\le H_m=G$ such that either $H_{i-1}$ is normal in $H_i$ or $\mathrm{\mid }{H}_i:H_{i-1}\mathrm{\mid }$ is a some prime $p$ and $p-1$ is not divisible by the $(t+1)$th powers of primes for every $i=1,\dots ,n$. In this work, properties of $\mathrm{K}$-${\mathbb{P}}_t$-subnormal subgroups and classes of groups with Sylow $\mathrm{K}$-${\mathbb{P}}_t$-subnormal subgroups are obtained.