\paragraph*{Context.} The endgame was to figure out whether $R^*$ is a \textbf{Mackey functor}, which means, among other things:
\begin{enumerate}
\item There is a transfer: a map $R^*(H) \rightarrow R^*(G)$, whenever $H$ is a subgroup of $G$. A good candidate for this would be the induction map, that turns a representation of $H$ into one of $G$.
\item \label{swanslemma} The transfer map satisfies Swan's Lemma: if H is an abelian $p$-Sylow of $G$, then the image of the restriction map $R^*(G) \rightarrow R^*(H)$ is the subring of $R^*(H)$ that is invariant under the map induced by the conjugation action of the normalizer $N_G(H)$ of $H$ in $G$. In equations:
\[
\Ima (res_H^G) = R^*(H)^{N_G(H)}
\]
\end{enumerate}
We started by testing whether Swan's lemma was true for the dihedral groups $D_p$, with $p$ an odd prime. It is, and we showed that in fact \textit{the induction is a graded ring homomorphism in that case}. However, it isn't in general, as the calculation of the graded ring $R^*(A_4,\CC)$ shows.
