Tracing projective modules over noncommutative orbifolds

For an action of a finite cyclic group $F$ on an $n$-dimensional noncommutative torus $A_\theta,$ we give sufficient conditions when the fundamental projective modules over $A_\theta$, which determine the range of the canonical trace on $A_\theta,$ extend to projective modules over the crossed product C*-algebra $A_\theta \rtimes F.$ Our results allow us to understand the range of the canonical trace on $A_\theta \rtimes F$, and determine it completely for several examples including the crossed products of 2-dimensional noncommutative tori with finite cyclic groups and the flip action of $\mathbb{Z}_2$ on any $n$-dimensional noncommutative torus. As an application, for the flip action of $\mathbb{Z}_2$ on a simple $n$-dimensional torus $A_\theta$, we determine the Morita equivalence class of $A_\theta \rtimes \mathbb{Z}_2,$ in terms of the Morita equivalence class of $A_\theta.$