@Z.M. No, I don't know the answer in this example.

@M.G. I did intentionally do algebraic-topology. This is of course not strictly speaking a question about algebraic topology, but these type of questions originated in this field, and many topologists (including myself) have thought a lot about these types of problems.

@ZhenLin This is the standard notation as far as I know, e.g., stacks.math.columbia.edu/tag/00AR or 4.3.7 of math.stanford.edu/~vakil/216blog/FOAGaug2922public.pdf, or in Hartshorne, etc.

BTW: For finite $G$, I believe the result you want follows from Corollary 4.2 of arxiv.org/pdf/2407.01507

The first spectral sequence can also be identified with the $K(n)$-local $E_n^{hH}$-Adams spectral sequence by ams.org/journals/tran/2005-357-01/S0002-9947-04-03394-X/…. It looks like the claimed isomoprhism might follow from $K(n)$-localization, and then some version of Morava's change of rings theorem.

There is a short argument in Remark 5.9 of math.uni-bielefeld.de/~hkrause/completion.pdf, which I believe shows that the ind-completion (in the triangulated sense) of the stable module category is not triangulated if the Sylow p-subgroup of $G$ is not cyclic.

Jeremy Rickard's very interesteing paper gives an answer to the current version of the question.

I was also worried about the multiplicative nature of the spectral sequence: it doens't cause problems that (at p = 2) K(n) does not admit a homotopy commutative ring structure?

If K_p(n) was a ring spectrum, this would follow from Corollary 4.5 of arxiv.org/pdf/2106.08669...I wonder if it is possible to make this type of argument work out

Current working link: web.archive.org/web/20130630192620/https://www.math.uwo.ca/…

The link doesn't work, but the thesis is available at: beautylabs.net/papers/index.html

One in fact has that an $E_n$-module is $K(n)$-local if and only if its homotopy groups are L-complete. A reference is Corollary 3.14 of arxiv.org/pdf/1311.7123.pdf.

Theorem 2.30 (and its proof) of arxiv.org/pdf/1507.06869.pdf are relevant