The Kirchberg-Phillips theorem from the mid 1990's was one of the early major breakthroughs in the Elliott classification programme for nuclear C*-algebras, giving a complete classification of Kirchberg algebras (in the UCT class) using K-theory. I will explain how this theorem can be generalised to classify amenable and suitably outer locally compact group actions on Kirchberg algebras, which as a special case confirms a conjecture of Izumi. This is joint work with Gábor Szabó.