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, (and really ultimately from ABGHR) 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 $\Delta(x)=x\otimes 1$.