This paper is a further study of finite Rokhlin dimension for actions of finite groups and the integers on C ∗ -algebras, intro- duced by the first author, Winter, and Zacharias. We extend the definition of finite Rokhlin dimension to the nonunital case. This def- inition behaves well with respect to extensions, and is sufficient to establish permanence of finite nuclear dimension and Z-absorption. We establish K-theoretic obstructions to the existence of actions of finite groups with finite Rokhlin dimension (in the commuting tower version). In particular, we show that there are no actions of any non- trivial finite group on the Jiang-Su algebra or on the Cuntz algebra O∞ with finite Rokhlin dimension in this sense. 2010 Mathematics Subject Classification: 46L55 1 The study of group actions on C ∗ -algebras, and their associated crossed prod- ucts, has always been a central research theme in operator algebras. One would like to identify properties of group actions which on the one hand occur com- monly and naturally enough to be of interest, and on the other hand are strong enough to be used to derive interesting properties of the action or of the crossed product. Examples of important properties for a group action meeting these criteria are the various forms of the Rokhlin property, which arose early on in the theory. See, for instance, (Izu01) and references therein for actions of Z and (Izu04a, Izu04b, Phi09, OP12) for the finite group case. The Rokhlin property for the single automorphism case is quite prevalent, and generic in some cases, forming a dense Gset in the automorphism group (see (HWZ15)). However, it