σ-properties of finite groups in polynomial time

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