  We prove that the natural homomorphism from Kirchberg's ideal-related
KK-theory, KKE(e, e'), with one specified ideal, into Hom_{\Lambda}
(\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all
extensions e and e' of separable, nuclear C*-algebras in the bootstrap category
N with the K-groups of the associated cyclic six term exact sequence being
finitely generated, having zero exponential map and with the K_{1}-groups of
the quotients being free abelian groups.
  This class includes all Cuntz-Krieger algebras with exactly one non-trivial
ideal. Combining our results with the results of Kirchberg, we classify
automorphisms of the stabilized purely infinite Cuntz-Krieger algebras with
exactly one non-trivial ideal modulo asymptotically unitary equivalence. We
also get a classification result modulo approximately unitary equivalence.
  The results in this paper also apply to certain graph algebras.
