Tornehave morphisms, II: The lifted Tornehave morphism and the dual of the Burnside functor

We introduce the lifted Tornehave morphism , an inflation Mackey morphism for finite groups, π being a set of primes, K the kernel of linearization, and B∗ the dual of the Burnside functor. For p-groups, is unique up to scalar multiples. It induces two morphisms of biset functors, one with a codomain associated with a subgroup of the Dade group, the other with a codomain associated with a quotient of the Burnside unit group.