$G$-kernels of Kirchberg algebras

A $G$-kernel is a group homomorphism from a group $G$ to the outer automorphism group of a C${}^{\ast }$-algebra. Inspired by recent work of Evington and Girón Pacheco in the stably finite case, we introduce a new invariant of a $G$-kernel using $K$-theory, and deduce several new constraints of the obstruction classes of $G$-kernels in the purely infinite case. We classify ${\mathbb{Z}}^n$-kernels for strongly self-absorbing Kirchberg algebras in the bootstrap category in terms of our new invariant and the Dadarlat-Pennig theory of continuous fields of strongly self-absorbing C${}^{\ast }$-algebras.