The previous paper showed that the G-surgery obstructions of G-normal maps lie in the Bak groups. That paper remarked that in even-dimensional cases, the G-surgery obstruction is invariant under suitable cobordisms. This paper presents cobordism invariance theorems for the G-surgery obstruction not only in even-dimensional cases but also in odd-dimensional ones. We prove Theorems B-D by detaching equivariant issues from the singular sets and then by using arguments of C. T. C. Wall in ordinary surgery theory. We still need, however, to argue carefully, especially in the odd-dimensional cases. Actually, this paper contains details which are skipped over in Wall's work. © 1990 Kluwer Academic Publishers.