So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it might be worth posting here, since if it's true, I think it'd be interesting.


Suppose I've got an inclusion of (at least) $\mathbb{E}\_2$-groups $j:H\hookrightarrow G$ such that I've got a fibration $H\overset{j}\to G\overset{q}\to G/H$, and I've got a Thom spectrum $Mf$ induced by a continuous map $f:G\to BGL\_1(\mathbb{S})$. From [Fridolin Roth's thesis](http://ediss.sub.uni-hamburg.de/volltexte/2009/4292/pdf/Dissertation_Fridolin_Roth.pdf), (and really ultimately from [ABGHR](http://arxiv.org/pdf/1403.4325.pdf)) we know that under nice circumstances (e.g. if $Mf$ is $H\mathbb{Z}$-oriented) then $Mf$ is a Hopf-Galois extension of the sphere spectrum with associated Hopf-algebra the spherical group ring $\mathbb{S}[G\_+]$. 


In particular, we have that the spectral sequence which computes the cofixed points of the coaction (in this case just the Thom diagonal) $Mf\to Mf\wedge \mathbb{S}[G\_+]$, which is just the BKSS of the cosimplicial spectrum computing that limit, converges to the homotopy of the sphere spectrum (it's just a sort of Adams spectral sequence). 


Now, the map $G\to G/H$ would seem to induce a coaction $$Mf\to Mf\wedge \mathbb{S}[G\_+]\to Mf\wedge\mathbb{S}[(G/H)\_+].$$ And thus one might ask what the cofixed points of this new coaction are. There should be at least a sort of "unit" map $\mathbb{S}\to Mf^{hco(G/H)\_+}$, and in an attempt to determine a sort of "intermediate" Hopf-Galois extension, one could hope to identify this object in between $\mathbb{S}$ and $Mf$. 


In the case that the morphism $G\to BGL\_1(\mathbb{S})$ is the trivial one, or in other words $Mf=\mathbb{S}[G\_+]$, one can identify this cofixed point object as $\mathbb{S}[H\_+]$ by noticing that the cosimplicial object defining the cofixed points is precisely the diagram defining the homotopy limit of the diagram $\ast\leftarrow \mathbb{S}[G\_+]\to \mathbb{S}[(G/H)\_+]$, or, in other words, the suspension spectrum of the homotopy fiber of the quotient map $G\to G/H$. This, and other vague considerations, have led me to believe that perhaps there is a general way of realizing the Thom spectrum $M(f\circ j)$ as the cofixed points of $Mf$ under the induced coaction of $\mathbb{S}[(G/H)\_+]$. I am concerned that there is some elementary consideration that would cause this to be true that I am just not seeing.


I have attempted to get at this thing by thinking about the bundles of spectra defining $Mf$ and $M(f\circ j)$, and using some kind of six-functor yoga, as well as just straight up computation, but haven't really gotten anywhere. Any thoughts would be dearly appreciated. 


One last remark: note that by cofixed points I *do not* mean homotopy orbits. In algebra, for a group $X$ which is coacted upon by a Hopf-algebra $C$, $\Delta:X\to X\otimes C$, the cofixed points of the coaction are the elements $x\in X$ such that $\Delta(x)=x\otimes 1$. 



Thom Spectra and Hopf-Galois Extensions of Ring Spectra
So this can definitely be done. It took me a while to figure out all the details, but in the end it's not so conceptually complex. 


The basic idea is that if you've got a fibration $F\overset{i}\to E\overset{p}\to B$ of connected $\mathbb{E}\_n$-spaces and a map $f:E\to BGL\_1(\mathbb{S})$ then you want to take the left Kan extension along $p$ as in the following (very poorly typeset) diagram:


$$\array{F\to & \!\!\!\!\!\!E\to &\!\!\!\!\!BGL\_1(\mathbb{S})\to Spectra\\ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\downarrow & \nearrow\\
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!B&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\longrightarrow} $$


This left Kan extension realizes the colimit of the morphism $B\to Spectra$ as $M(f\circ i)/\Omega B$, which can be thought of as $M(f\circ i)/(\Omega E/\Omega F)$ or even as $(\mathbb{S}/\Omega F)/(\Omega E/\Omega F)$. But we can also (following the answer to [this question](https://mathoverflow.net/questions/218866/iterated-homotopy-quotient)) recognize this as the Kan extension along $E\to \ast$, giving $M(f\circ i)/\Omega B\simeq Mf$. Moreover, this can be shown to produce $Mf$ as a Thom spectrum over $M(f\circ i)$, which immediately gives you the $B$-coaction on $Mf$ and the torsor condition of Hopf-Galois extensions $Mf\otimes\_{M(f\circ i)}Mf\simeq Mf\otimes \mathbb{S}[B]$.


To get that $M(f\circ i)$ can be recovered as the primitives of the $B$-coaction, notice that if everything is $H\mathbb{Z}$-oriented then in homology the descent, or primitives, spectral sequence computing the cotensor product $Mf\Box\_B\mathbb{S}$ is equivalent to the Eilenberg-Moore spectral sequence computing $E\Box\_B\ast\simeq E\times\_B \ast\simeq F$. In other words (and this is very rough), in homology, both spectral sequences converge to the homology of $F$, which (if we assume $H\mathbb{Z}$-orientation) is exactly the homology of $M(f\circ i)\simeq \mathbb{S}/\Omega F$. 


I've worked out the details in a preprint [here](http://arxiv.org/pdf/1601.04123.pdf). 


