Let $G$ be a finite group and $X$ a pointed $G$-space. The assignment $G/H\to \pi_n(X^H)$ should define a Mackey functor. I am trying to figure out what the transfers and restrictions are.
If $H\subset K$, then $X^K\subset X^H$ which induces a map $\pi_n(X^K)\to\pi_n(X^H)$ that I would tentatively called the restriction $Res_H^K$ (thanks for pointing out the mistake). Similarly, left multiplication induces a map $\pi_n(X^H)\to \pi_n(X^{gHg^{-1}})$, that I would call $C_g$.
Unfortunately, I am not sure that these are the right definitions. There is a similar question for $G$-spectra and the construction for those maps seems pretty elaborate and requires of the ''Wirthmüller isomorphism''. I was hoping my natural definitions are correct (please tell me if they are not) and you can explain how $Tr_H^K:\pi_n(X^H)\to\pi_n(X^K)$ is defined or provide a reference with a detailed exposition.
The transfer is a stable phenomenon and cannot exist at the space level. To see why, let $G$ be the cyclic group of order $2$ with generator $c$ and let $X=S^1\wedge EG_+$, where $G$ acts trivially on $S^1$. Then $G$ acts trivially on $\pi_1(X)\cong \pi_1(S^1)=\mathbb{Z}$.
If the homotopy groups were to have the structure of a Mackey functor, then the double coset formula would say that $\mathrm{Res}\circ \mathrm{Tr}=1+c$, which is multiplication by $2$ on $\pi_1(X)$, since $G$ acts trivially on $\pi_1(X)$. But $X^G$ is a point, hence $\pi_1(X^G)$ is trivial and we arrive at a contradiction.