  Let $K$ be a finite group and let $G$ be a finite group acting on $K$ by
automorphisms. In this paper we study two different but intimately related
subjects: on the one side we classify all possible multiplicative and
associative structures with which one can endow the twisted $G$-equivariant
K-theory of $K$, and on the other, we classify all possible monoidal structures
with which one can endow the category of twisted and $G$-equivariant bundles
over $K$. We achieve this classification by encoding the relevant information
in the cochains of a sub double complex of the double bar resolution associated
to the semi-direct product $K \rtimes G$; we use known calculations of the
cohomology of $K$, $G$ and $K \rtimes G$ to produce concrete examples of our
classification.
  In the case in which $K=G$ and $G$ acts by conjugation, the multiplication
map $G \rtimes G \to G$ is a homomorphism of groups and we define a shuffle
homomorphism which realizes this map at the homological level. We show that the
categorical information that defines the Twisted Drinfeld Double can be
realized as the dual of the shuffle homomorphism applied to any 3-cocycle of
$G$. We use the pullback of the multiplication map in cohomology to classify
the possible ring structures that the Grothendieck ring of representations of
the Twisted Drinfeld Double may have, and we include concrete examples of this
procedure.
