Morita equivalence classes for crossed product of rational rotation algebras

We study the Morita equivalence classes of crossed products of rotation algebras A_θ, where θ is a rational number, by finite and infinite cyclic subgroups of SL(2, Z). We show that for any such subgroup F, the crossed products A_θ F and Aθ' F are strongly Morita equivalent, where both θ and θ' are rational. Combined with previous results for irrational values of θ, our result provides a complete classification of the crossed products A_θ F up to Morita equivalence.