The first homology of the group of equivariant diffeomorphisms and its applications

Let V be a representation space of a finite group G. We determine the group structure of the first homology of the equivariant diffeomorphism group of V. Then we can apply it to the calculation of the first homology of the corresponding automorphism groups of smooth orbifolds, compact Hausdorff foliations, codimension one or two compact foliations and the locally free S 1-actions on 3-manifolds.