of (compactly-supported) *Heisenberg* pseudodifferential operators. The latter is a modification of the classical algebra of pseudodifferential operators, in which the vector fields tangent to the leaves of the foliation are of order 1, while the transverse vector fields are of order $\leq 2$. Let $G\subset {\mathrm{Diff}}(M)$ denote a discrete group of diffeomorphisms on $M$ mapping leaves to leaves. Then $G$ naturally acts on the algebra $\Psi_{H,c}(M)$ by automorphisms, and the crossed-product $\Psi_{H,c}(M)\rtimes G$ is defined. We write any element $P \in \Psi_{H,c}(M) \rtimes G$ as a finite sum $$P = \sum_{g \in G} P_g \otimes U_g\ .$$ Remark that $P$ is naturally represented as a linear operator $\sum P_g \circ U_g : C^{\infty}_c(M) \longrightarrow {C^{\infty}}_c(M)$, where $U_g$ acts as the shift operator $U_g(f)(x) = f(x \cdot g)$, for every $f\in {C^{\infty}_c}(M)$ and $x \in M$. Thus $P$ is far from being pseudodifferential in general, and belongs to the larger class of Fourier integral operators. At least at the algebraic level (see however the work of Savin and Sternin [@SS]), the index theory of such operators amounts to describe the $K$-theory/cyclic homology excision maps associated to the short exact sequence of algebras $$0 \longrightarrow \Psi^{-\infty}_c(M) \rtimes G \longrightarrow \Psi_{H,c}^{0}(M) \rtimes G 