Graded character rings, Mackey functors and Tambara functors

Let $G$ be a finite group and $\mathbb{K}$ a field of characteristic zero. the ring $R_\mathbb{K}(G)$ of virtual characters of $G$ over $\mathbb{K}$ is naturally endowed with a so-called Grothendieck filtration, with associated graded ring $R^*_\mathbb{K}(G)$. Restriction of representations to any $H\leq G$ induces a homomorphism $R^*_\mathbb{K}(G) \to R^*_\mathbb{K}(H)$. We show that, when $G$ is abelian, induction of representations preserves the filtration, so $R^*_\mathbb{C}(-)$ is a Mackey functor; in the general case, we propose a modified filtration which turns $R^*_\mathbb{K}(-)$ into a Mackey functor. We then turn to tensor induction of representations, and show that in the abelian case $R^*_\mathbb{C}(-)$ is a Tambara functor.