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.