Let $G$ be a compact Lie group. We can define $\mathfrak{F}G$ to be the collection of conjugacy classes of closed subgroups of $G$ whose Weyl group is finite, a bi-invariant metric on $G$ induces a topology on this set. It is a result by tom Dieck that there is a rational isomorphism \begin{equation} \pi_0^{G}(S^0)\cong C(\mathfrak{F}, \mathbb{Q}) \end{equation} sending $f$ to the function assigning $K$ to $\text{deg} \ \phi^K(f)$. Where $\phi^K$ denotes the usual geometric fixed points and $C(-,-)$ the set of continuous functions with $\mathbb{Q}$ having the discrete topology.
In the case of $G=O(2)$ the closed subgroups with finite Weyl groups are $O(2)$, $SO(2)\cong S^1$ and the dihedral subgroups $D_{2n}$. It is possible to show that in $\mathfrak{F}O(2)$ the circle is an isolated point, and so are the conjugacy classes of the $D_{2n}$'s, instead $O(2)$ is a limit point of the dihedral subgroups. By ease of notation we set $\mathcal{D}$ to be the subspace of $\mathcal{F}O(2)$ not including $SO(2)$.
This implies that the identity of the sphere splits as the sum of two idempotents $e_{\mathcal{C}}$ and $e_\mathcal{D}$, where $e_{\mathcal{C}}$ corresponds to the characteristic function of $SO(2)$ while $e_\mathcal{D}$ to the characteristic function of $\mathcal{D}$. This provides a decomposition of the rational $O(2)$-equivariant stable homotopy category which is the first step in its analysis. This is well documented in the literature, see for example "Rational $O(2)$-equivariant cohomology theories" by Greenlees or the more recent "Rational $O(2)$-Equivariant Spectra" by Barnes.
The part associated to the idempotent $e_{\mathcal{C}}$ is easy to understand: it is equivalent to the stable category of rational $S^1$-spectra together with an action of $O(2)/SO(2)\cong C_2$. My interest is on the dihedral part associated to $e_{\mathcal{D}}$, thus from now for any rational $O(2)$-spectrum $X$ I will implicitly mean the dihedral part $e_{\mathcal{D}}X$.
My question is the following: let $\mathcal{F}$ be the the family of subgroups of $O(2)$ generated by the finite dihedral subgroups. Then it should hold that \begin{equation} [E\mathcal{F}_+, E\mathcal{F}_+]\cong C(\mathcal{D}\setminus \{ O(2)\},\mathbb{Q}). \end{equation} I find this reasonable: we have $\phi^KE\mathcal{F}_+\cong \phi^KS^0$ for all the subgroups $K$ isomorphic to $D_{2n}$ and $\phi^HE\mathcal{F}=0$ if $H \not \in \mathcal{F}$, thus this claim states that $E\mathcal{F}_+$ sees the part of $\pi_0(S^0)$ individuated by the geometric fixed points with respect to dihedral groups.
But I did not manage to find a proof, nor come come up with one myself. I found this claim in notes without a reference so I do not know where to look for one, but since the $O(2)$-equivariant spectra were studied plenty in the literature I believe there should be a paper or book where this isomorphism is proved.
Can you provide a reference or give a summary of the proof?
