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 relation to their paper.

Let $G$ be a finite group.

In their paper [Some Remarks On the Structure of Mackey functors][1] , 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?

  [1]: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&ved=0CEUQFjAE&url=http://www.ams.org/proc/1992-115-01/S0002-9939-1992-1076574-2/S0002-9939-1992-1076574-2.pdf&ei=ZaVSUPP5M4f28wT15IC4DQ&usg=AFQjCNFuJ21qi2OIXYT18tuVtjnzDoxFeg&sig2=PhtFfjAqaoK5ytWaVyeJvw