A G-kernel is a group homomorphism from a group G into the outer automorphism group of a C*-algebra. In contract to the case of injective von Neumann algebras where the classfication theory of G-kernels is well-established, almost nothing had been known in the C*-algebra case until Evington-Girón Pacheco recently showed that the third cohomology obstruction of a G-kernel always vanishes for the Jiang-Su algebra.

We introduce a new K-theoretical invariant for G-kernels of C*-algebras, and discuss a classification problem of G-kernels in the case of Kirchbergs.