Finite groups with σ-Frobenius condition for non-normal σ-primary subgroups

Let sigma = {sigma(i) vertical bar i is an element of I} be a partition of the set P of all primes and C a finite group. A group is said to be sigma-primary if it is a finite sigma(i)-group for some i. We say that a sigma(i)-subgroup H of G satisfies the sigma-Frobenius condition in G if N-G(H)/C-G(H) is a sigma(i)-group. In this paper, we determine the structure of finite groups in which every non-normal sigma-primary subgroup satisfies the sigma-Frobenius condition.