We introduce the outer conjugacy invariants , for cocycle actions of discrete groups G on type factors N, as the set of real numbers for which the amplification of can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly and the fundamental group of , , in the case G has infinite normal subgroups with the relative property (T) (e.g., when G itself has the property (T) of Kazhdan) and is an action of G on the hyperfinite factor by Connes–Størmer Bernoulli shifts of weights . Thus, and coincide with the multiplicative subgroup S of generated by the ratios , while if (i.e. when all weights are equal), and otherwise. In fact, we calculate all the “1-cohomology picture” of and classify the actions in terms of their weights . In particular, we show that any 1-cocycle for vanishes, modulo scalars, and that two such actions are cocycle conjugate iff they are conjugate. Also, any cocycle action obtained by reducing a Bernoulli action of a group G as above on to the algebra pNp, for p a projection in N, , cannot be perturbed to a genuine action.