Borel complexity and automorphisms of C*-algebras

We show that if A   is ZZ, O2O2, O∞O∞, a UHF algebra of infinite type, or the tensor product of a UHF algebra of infinite type and O∞O∞, then the conjugation action Aut(A)↷Aut(A)Aut(A)↷Aut(A) is generically turbulent for the point-norm topology. We moreover prove that if A   is either (i) a separable C*-algebra which is stable under tensoring with ZZ or KK, or (ii) a separable II1II1 factor which is McDuff or a free product of II1II1 factors, then the automorphisms of A are not classifiable up to conjugacy by countable structures.