  G.W. Mackey's celebrated obstruction theory for projective representations of
locally compact groups was remarkably generalized by J. M. G. Fell and R. S.
Doran to the wide area of saturated Banach *-algebraic bundles.
  Analogous obstruction is suggested here for discrete group graded algebras
which are not necessarily saturated, i.e. strongly graded in the discrete
context. The obstruction is a map assigning a certain second cohomology class
to every equivariance class of absolutely simple graded modules.
  The set of equivariance classes of such modules is equipped with an
appropriate multiplication, namely a graded product, such that the obstruction
map is a homomorphism of abelian monoids. Graded products, essentially arising
as pull-backs of bundles, admit many nice properties, including a way to twist
graded algebras and their graded modules.
  The obstruction class turns out to determine the fine part that appears in
the Bahturin-Zaicev-Sehgal decomposition, i.e. the graded Artin-Wedderburn
theorem for graded simple algebras which are graded Artinian, in case where the
base algebras (i.e. the unit fiber algebras) are finite-dimensional over
algebraically closed fields.
