Currently I'm working on the following version of the AHSS $$ E^2_{pq}\cong H_p(M\eta; MSpin_q(\ast))\Rightarrow MSpin_{p+q}(M\eta)$$ where $\eta \colon B \to BSO$ is a stable vector bundle, and $M\eta$ denotes its Thom Spectrum.

My experience with the AHSS is "graduate level", therefore most properties and, let's say, tricks I'm aware of it is for the simple version with $E^2_{pq}(X)\Rightarrow h_*(X)$ where $X$ is a CW space, NOT a spectrum.

So these are the questions I'm looking for an answer (and a reference for it)

- Third differentials for $ E^2_{pq}\cong H_p(M\eta; MSpin_q(\ast))\Rightarrow MSpin_{p+q}(M\eta)$.

I was able to find a reference for the second differentials (https://math.berkeley.edu/~teichner/Papers/Signature.pdf) Prop 1 page $750$, but nothing for the third differentials. In my specific case, there is a possibly non-trivial $d_3\colon E^3_{5,0}\to E^3_{2,2}$. Asking around it seems that it's general knowledge that the $d_3$ starting from the $0$th row is a secondary cohomology operation associated to $Sq^2Sq^2=0$. I'm not able to find any reference for this, and my little knowledge in secondary cohomology operations prevents me from proving it on my own.

- Edge homomorphisms: Are there some characterisations for the edge homomorphisms in this case?

I'm aware that the characterisation can't be something easy, but if there is something it might prove useful sometimes.

- Is there some multiplicative structure on the cohomological version?

I'm referring to something on the lines of Kochman page 34 def. $2.2.1$, in particular differentials are derivations. I think there should be, but I'm looking for some opinion from someone more experienced than me.

- Is there a pairing between the AHSS's for $MSpin_*(M\eta)$ and $MSpin^*(M\eta)$ like in the case for a finite CW complex $X$?

I'm referring to a pairing analogue to the one explained in Kochman page 129 Proposition $4.2.10$. In particular I'd like to have some kind of relationship between the differential in the homology version and in the cohomology version, since my idea is to try figure out the cohomological one (maybe using some kind of multiplicative structure) and infer something about the homological ones.

Thanks in advance for your help.