(Non)-equivariant equivalence in $G$-spectra

Question

In HHR, an important part is the periodicity theorem. For proving the theorem, they invert a carefully defined class $D \in \pi^{C_8}_{19\rho_8}(N^8_2MU_{\mathbb{R}})$ and they can find an element in $x \in \pi_{256}^{C_8}D^{-1}N^8_2MU_{\mathbb{R}}$, when we forget the $C_8$ equivariant structure, gives a non-equivariant equivalence from $\Sigma^{256}D^{-1}N^8_2MU_{\mathbb{R}}$ to $D^{-1}N^8_2MU_{\mathbb{R}}$. This non-equivariant equivalence is sufficient for them to show that the homotopy fixed point spectrum has the same periodicity.

However, in a following paper, they claim in several places that the periodicity theorem they proved in HHR, shall gives us an equivariant equivalence, which is stronger than the original statement I have seen in the Kervaire invariant one paper. So here is my question:

Given a finite group $G$ and a $G$-spectrum $X$, if we have an $G$-equivariant self-map $f:\Sigma^mX \rightarrow X$, which induces an equivalence of the underlying non-equivariant spectrum, will $f$ automatically be an $G$-equivariant equivalence? If not, is there any condition we can apply to $G$ or $X$ to make this statement true?

If the answer is positive, the strengthened periodicity theorem will give us a computational advantage in the sense that normally slice spectral sequence in positive and negative dimension looks vastly different, and when come to resolve extension problems in $E_\infty$ page, an equivariant periodicity will induces isomorphism as Mackey functors, rather than Abelian groups. It will automatically resolve most of the extension problem by simple comparison.

I have asked all people I could reach in real life, but I do not have a satisfying answer yet.

Any answer or hint is greatly appreciated.

Answer 1

The condition required on $X$ which makes this work is that $X$ is cofree (Definition 10.1 in the linked paper): the map $X \to F(EG_+,X)$ is an equivalence. For any equivariant map $X \to Y$ of cofree $G$-spectra which is an equivalence on the underlying spectra, the resulting maps $X^H \to Y^H$ of fixed-point sets are equivalent to the weak equivalence of homotopy fixed-point spectra $X^{hH} \to Y^{hH}$. This makes $X \to Y$ into a genuine equivalence.

The fact that $D^{-1} N_2^8 MU_{\Bbb R}$ is cofree is an important result in their paper (Theorem 10.8).