Kernels, inflations, evaluations, and imprimitivity of Mackey functors

Let M be a Mackey functor for a finite group G. By the kernel of M we mean the largest normal subgroup N of G such that M can be inflated from a Mackey functor for G/N. We first study kernels of Mackey functors, and (relative) projectivity of inflated Mackey functors. For a normal subgroup N of G, denoting by the projective cover of a simple Mackey functor for G of the form we next try to answer the question: how are the Mackey functors and related? We then study imprimitive Mackey functors by which we mean Mackey functors for G induced from Mackey functors for proper subgroups of G. We obtain some results about imprimitive Mackey functors of the form , including a Mackey functor version of Fongʹs theorem on induced modules of modular group algebras of p-solvable groups. Aiming to characterize subgroups H of G for which the module is the projective cover of the simple -module V where the coefficient ring is a field, we finally study evaluations of Mackey functors.