  Counterexamples to classification of purely infinite, nuclear, separable
C*-algebras (in the ideal-related bootstrap class) and with primitive ideal
space X using ideal-related K-theory occur for infinitely many finite primitive
ideal spaces X, the smallest of which having four points. Ideal-related
K-theory is known to be strongly complete for such C*-algebras if they have
real rank zero and X has at most four points for all but two exceptional
spaces: the pseudo-circle and the diamond space. In this article, we close
these two remaining cases. We show that ideal-related K-theory is strongly
complete for real rank zero, purely infinite, nuclear, separable C*-algebras
that have the pseudo-circle as primitive ideal space. In the opposite
direction, we construct a Cuntz-Krieger algebra with the diamond space as its
primitive ideal space for which an automorphism on ideal-related K-theory does
not lift.
