We discuss conditions for the Aubin property of solutions to perturbed cone constrained programs, by using and refining results given in Klatte-Kummer "Nonsmooth Equations in Optimization", Kluwer, 2002. In particular, we show that constraint nondegeneracy and hence uniqueness of the multiplier is necessary for the Aubin property of the critical point map. Moreover, we give conditions under which the critical point map has the Aubin property if and only if it is locally single-valued and Lipschitz
