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 P_{H, V}^G the projective cover of a simple Mackey functor for G of the form S_{H, V}^G we next try to answer the question: how are the Mackey functors P_{H / N, V}^{G / N} and P_{H, V}^G 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 P_{H, V}^G, including a Mackey functor version of Fong's theorem on induced modules of modular group algebras of p-solvable groups. Aiming to characterize subgroups H of G for which the module P_{H, V}^G (H) is the projective cover of the simple K over(N, -)_G (H)-module V where the coefficient ring K is a field, we finally study evaluations of Mackey functors.