On the partial $\mathcal{L}$-$\mathrm{\Pi }$-property of subgroups of finite groups

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the partial $\mathcal{L}$-$\mathrm{\Pi }$-property in $G$ if $H⊴G$, or if $\mathrm{\mid }G\mathrm{/}K:{\mathrm{N}}_{G\mathrm{/}K}(HK\mathrm{/}K)\mathrm{\mid }$ is a $\pi (HK\mathrm{/}K)$-number for any $G$-chief factor of type $H^G\mathrm{/}K$ with $H_G\le K$. In this paper, we investigate the structure of finite groups under the assumption that some subgroups of prime power order satisfy the partial $\mathcal{L}$-$\mathrm{\Pi }$-property.