[2309.03441] $G$-kernels of C$*$-algebras
A $G$-kernel is a group homomorphism from a group $G$ to the outer
automorphism group of a C$^*$-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$^*$-algebras.