Mackey functors and functorial extensions

Given a finite group X and a normal subgroup G of X, we show that any Mackey functor M for X induces another Mackey functor M˜ for X associated to G. We then consider the question, whether there exists a map M˜→M extending elements from M(G) to M(X) and compatible with the restriction maps. In the case that the order of G and the index of G in X are relatively prime, we give a sufficient condition for the existence of such a map, using canonical induction formulae.