For a discrete group G and a C*-algebra A a G-kernel on A is a homomorphism from G to outer automorphism group of A. To any G-kernel, one can associate a cohomological obstruction to lifting it to a group action on A. In a recent work with Evington and subsequent work of Izumi, new cohomological invariants of G-kernels were introduced. These invariants place restrictions to the possible values of lifting obstructions on C*-algebras. In this talk, I will discuss how these obstructions can be interpreted as cohomology with coefficients in crossed modules and the benefits of this framework. This is joint work with Masaki Izumi and Ulrich Pennig.
