  A Mackey type decomposition for group actions on abelian categories is
described. This allows us to define new Mackey functors which associates to any
subgroup the $K$-theory of the corresponding equivariantized abelian category.
In the case of an action by tensor autoequivalences the Mackey functor at the
level of Grothendieck rings has a Green functor structure. As an application we
give a description of the Grothendieck rings of equivariantized fusion
categories under group actions by tensor autoequivalences on graded fusion
categories.
