In [12] we introduced and studied properties ( gab ) and ( gaw ), which are extensions to the context of B-Fredholm theory, of properties ( ab ) and ( aw ) respectively introduced also in [12]. In this paper we continue the study of these properties and we consider their stability under commuting finite rank, compact and nilpotent perturbations. Among other results, we prove that if T is a bounded linear operator acting on a Banach space X , then T possesses property ( gaw ) if and only if T satisfies generalized Weyl's theorem and E ( T ) = Ea ( T ). We prove also that if T possesses property ab or property ( aw ) or property ( gaw ) respectively, and N is a nilpotent operator commuting with T , then T+N possesses property ab or property aw or property ( gaw ) respectively. The same result holds for property ( gab ) in the case of a-polaroid operators.