Group graded algebras and almost polynomial growth

Let F be a field of characteristic 0, G a finite abelian group and A a G-graded algebra. We prove that A generates a variety of G-graded algebras of almost polynomial growth if and only if A has the same graded identities as one of the following algebras:(1)FCp, the group algebra of a cyclic group of order p, where p is a prime number and p||G|;(2), the algebra of 2×2 upper triangular matrices over F endowed with an elementary G-grading;(3)E, the infinite dimensional Grassmann algebra with trivial G-grading;(4)in case 2||G|, EZ2, the Grassmann algebra with canonical Z2-grading.