Let F be an algebraically closed field of characteristic zero and let G be a finite group. In this paper we will show that the asymptotics of c G n (A), the Ggraded codimension sequence of a finite dimensional G-simple F -algebra A, has the form αn 1−dim F Ae 2 (dim F A) n (as conjectured by E.Aljadeff, D.Haile and M. Natapov), where α is some positive real number and A e denotes the identity component of A. In the special case where A is the algebra of m×m matrices with an arbitrary elementary G-grading we succeeded in calculating the constant α explicitly.