  We introduce a new construction, the isotropy groupoid, to organize the orbit
data for split $\Gamma$-spaces. We show that equivariant principal $G$-bundles
over split $\Gamma$-CW complexes $X$ can be effectively classified by means of
representations of their isotropy groupoids. For instance, if the quotient
complex $A=\Gamma\backslash X$ is a graph, with all edge stabilizers toral
subgroups of $\Gamma$, we obtain a purely combinatorial classification of
bundles with structural group $G$ a compact connected Lie group. If $G$ is
abelian, our approach gives combinatorial and geometric descriptions of some
results of Lashof-May-Segal and Goresky-Kottwitz-MacPherson.
