Let R be a commutative ring and G a finite abelian group. In [8], Long developed a Brauer group theory for G-dimodule algebras (i.e., algebras with a compatible G-grading and G-action) and constructed BD(R, G), the Brauer group of G-Azumaya algebras. Within BD(R, G) lies B(R, G), the set of classes of algebras which are R-Azumaya (i.e., central separable) as well as G-Azum$ya. B(R, G) is not always a group; we show that if every cocyle in H2(G, U(R)) is abelian, then it is. When B(R, G) is a group, we call it the Brauer group of central separable G-Azumaya algebras. If R is connected and Pit,(R) = 0 where m is the exponent of G, and if every cocycle in H2(G, U(R)) is abelian, then we show that there is a short exact sequence 1 + (BC(R, G)/B(R)) x (BM(R, G)/B(R)) -+ B(R, G)/B(R) -+ Aut(G) + 1, where B(R) is the usual Brauer group of R, BM(R, G) is the Brauer group of G-module algebras and BC(R, G) is the Brauer group of G-comodule algebras (cf. [S]). If either BM(R, G)/B(R) or BC(R, G)/B(R) is trivial, then the sequence splits. Using the above, we are able to describe BD(Z, G) for any cyclic G, and BD(IW, G) for any cyclic G of odd order. Our sequence provides a generalization of the results [9, Theorem 5.91 and [II, Theorem 4.41 and should be compared to the sequence in [6, Theorem 5.21 obtained under the assumptions that the order of G is a unit in R and R contains a primitive mth root of unity.