$\sigma$-properties of finite groups in polynomial time

Let $H,K$ be subgroups of the permutation group $G$ of degree $n$ with $K⊴G$ and $\sigma$ be a partition of the set of all different prime divisors of $\mathrm{\mid }G\mathrm{/}K\mathrm{\mid }$. We prove that in polynomial time (in $n$) one can check $G\mathrm{/}K$ for $\sigma$-nilpotency and $\sigma$-solubility; $H\mathrm{/}K$ for $\sigma$-subnormality and $\sigma$-$p$-permutability in $G\mathrm{/}K$. Moreover one can find the least partition $\sigma$ of $\pi (G\mathrm{/}K)$ for which $G\mathrm{/}K$ is $\sigma$-nilpotent. Also one can find the least partition $\sigma$ of $\pi (G\mathrm{/}K)$ for which $H\mathrm{/}K$ is $\sigma$-$p$-permutable in $G\mathrm{/}K$.