  In the first part of this paper, we propose a uniform interpretation of
characteristic classes as obstructions to the reduction of the structure group
and to the existence of an equivariant extension of a certain homomorphism
defined a priori only on a single fiber of the bundle. By plugging in the
correct parameters, we recover several classical theorems. Afterwards, we
define a family of invariants of principal bundles that detect the number of
group reductions that a principal bundle admits. We prove that they fit into a
long exact sequence of abelian groups, together with the cohomology of the base
space and the cohomology of the classifying space of the structure group.
