  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 $A_\theta.$
