Let $G$ be a finite group ad let $X$ be a $G$-space. Consider the following Mackey functor, that I will denote by $\underline{\pi}_n$: $G/H\mapsto \pi_n^H(X)$, where $\pi_n^H(X)$ refers to the stable homotopy group of $(G/H)_+\wedge X$.
If $K\subset H$, then the canonical map $p:G/K\to G/H$ induces the restriction map $Res_K^H$. Conjugation should also be induced by the obvious map $C_g$ as $\pi_n^H(C_g)$ or something like that.
The transfer map seems to be the tricky one. A brief explanation is given in Peter May, p. 84, as follows:
For $H/K$, there is a transfer map $t(H/K):S^n\to (H/K)_+\wedge S^n$ constructed using the Pontrjagin-Thom map. I will take this as a fact. I am more intrigued about the map $t(p)$, defined a using the above $t(H/K)$ as follows: $$ t(p):(G/H)_+\wedge S^n \cong G_+\wedge_H S^n \overset{t(H/K)}{\longrightarrow} G_+\wedge_H(H/K)_+\wedge S^n \cong (G/K)_+\wedge S^n. $$
My question is:
Is the trasnfer $Tr^H_K$ obtained by pushing forward $t(\pi)$, that is, $Tr_K^H$ is the induced map $$ t(\pi)_*:\{S^n,(G/H)_+\wedge X\}_G \longrightarrow \{S^n,(G/K)_+\wedge X\}_G ? $$
(If $X$ and $Y$ are finite $G$-CW complexes, I denote by $\{X,Y\}_G$ the colimit $\varinjlim_k [S^k\wedge X,S^k\wedge Y]_G$.)
Thank you.
PS: Notice the big effort I have made to avoid spectra.