In [HHR](http://www.math.rochester.edu/people/faculty/doug/mypapers/Hill_Hopkins_Ravenel.pdf), 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](http://arxiv.org/pdf/1502.07611v1.pdf), 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.



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


