  In this paper, for a compact Lie group action,we prove the anomaly formula
and the functoriality of the equivariant Bismut-Cheeger eta forms with
perturbation operators when the equivariant family index vanishes. In order to
prove them, we extend the Melrose-Piazza spectral section and its main
properties to the equivariant case and introduce the equivariant version of the
Dai-Zhang higher spectral flow for arbitrary dimensional fibers.Using these
results, we construct a new analytic model of the equivariant differential
K-theory for compact manifolds when the group action has finite stabilizers
only,which modifies the Bunck-Schick model of the differential K-theory. This
model could also be regarded as an analytic model of the differential K-theory
for compact orbifolds. Especially, we answer a question proposed by Bunke and
Schick about the well-definedness of the push-forward map.
