In the special case, where $A$ is the group vNa algebra of a type 1 group $H$, and $\alpha$ acts on $H$ via automorphism, then the extremal states on $A \rtimes_\alpha G$ correspond roughly to irreducible representations of $H \rtimes_\alpha G$ (assume that for instance that $H$ is compact), and these can be computed via the Mackey machine (Look at Mackey's paper "Group extensions of locally compact groups"). I would guess that Mark Rieffel has extended this to more general settings, but I am not very knowledgable in this area.