... in fact $F$ is self-adjoint!

@Johannes: Sorry, you are right, that was a very stupid mistake. In fact, the most convenient picture of $K^{p,q}(X)$ to use here is the Fredholm one given in Karoubi's paper Algèbres de Clifford et operateurs de Fredholm, CRAS, t. 267, p. 305 (1968) (Section II): $K^{p,q}(X)$ is generated by pairs $(\mathcal{E},F)$ where $\mathcal{E}\longrightarrow X$ is a graded Hilbert bundle equipped with a $Cl^{p,q}$-module structure, and $F\in C(X,Fred_\mathcal{E})$ is a Fredholm section of degree $1$ anticommuting with the generators of $Cl^{p,q}$. Such a pair is but a $KK$-class.

@Mikhail: A groupoid isomorphism from $A\rightrightarrows X$ to $G\ltimes X\rightrightarrows X$ induces a $\Gamma$-action on the group bundle $G\ltimes X$. One obtains the $\Gamma$-action on $G$ by identifying $G$ with the fibres $(G\ltimes X)_x=(G\ltimes X)^x=(G\ltimes X)^x_x$.

Thanks, Carnahan, for this very interesting answer which, combined with Ronnie's intuitive bi-groupoid view point, helped me so much.

Here are some places in Europe with very good and reasonably big research groups in operator algebras and non-commutative geometry: - Paris Jussieu (G. Skandalis,M. Hilsum, A. Zuk, E. Blanchard, etc.); - Orléans (V. Lafforgue, J. Renault, etc.); - Metz (J.-L. Tu, M. Benameur, H. Oyono, etc.); - Marseille (Puschnig, etc.); - Muenster (J. Cuntz, S. Echterhoff, etc.); - Goettingen (R. Meyer,etc.); - Coppenhagen (R. Nest, Kirchberg, etc.).