This is somewhat related to a question that I asked on Math.SE but, sadly, received no response. I apologize ahead of time if this is not appropriate for MO. Feel free to vote to close if this is the case.

I really have two (distinct) questions. The first is regarding a paper by Greenlees and May and the second is more of a "big-picture" question with no explicit relationship to their paper.

Let $G$ be a finite group.

In their paper Some Remarks On the Structure of Mackey functors , Greenlees and May define the functor:

$R: GMod \rightarrow M[G]$ where $GMod$ is the category of finite left $G$-modules and $M[G]$ is the category of $G$ Mackey functors by:

$RV(G/H) = V^H$ where $V$ is a $G$ module and $V^H$ is the $H$ fixed point set of $V$.

In their main theorem(Thm. 12) they consider the map $\eta: M \rightarrow RM(G/H_{j,k})$.

Question 1: In Theorem 12, why are $coker(\eta)$ and $ker(\eta)$ in $\mathcal{A}$? Unfortunately I don't see how this clearly follows from the induction hypothesis at the moment.

Question 2: In general, What are some examples of added benefits (aside from additional structure) that one obtains when it is known that you have a Green or Tambara functor rather than just a Mackey functor?