Some clarifications about the Secondary Cohomology Operation associated to $Sq^2\circ Sq^2=0$

Question

As explained in the title, I'm looking for some clarifications about the secondary cohomology operation associated to the relation $Sq^2\circ Sq^2=0$. I've just started reading the relevant chapter in the book of "Harper, Secondary Cohomology Operations" but there are some unclear passages.

1. Why the relation $Sq^2\circ Sq^2=0$ should hold? I mean, I know that $Sq^2$ can be identified with the differential in some instances of the AHSS, therefore behind a differential it squares to zero. What I'm interested is in a more direct proof
2. Following the axiomatic approach one says that the sec. coh. op $\Phi$ is associated to the relation $Sq^2\circ Sq^2=0$ if we are provided maps representing $Sq^2$'s and a nullhomotopy of the composition: $$ K(\mathbb{Z}_2,1) \xrightarrow{Sq^2} K(\mathbb{Z}_2,3) \xrightarrow{Sq^2} K(\mathbb{Z}_2,5)$$ where I fixed the indices reflecting the case I'm interested in. Let us call the nullhomotopy $H$. Now as remainded in the Omnibus Theorem (Thm $4.2.4$ page 98), there is a dependency of $\Phi$ on the choice of such $H$, but this seems to be not mentioned in a lot of similar examples. Since this $\Phi$ is supposed to represent the third differential of the AHSS, I suspect that there is no such dependency, I've no idea why though.
3. Are there some clever ways to do the computations of the image of elements under $\Phi$ or one has just to follow the construction of the secondary operation and try chasing the elements around it?

Thanks in advance for all the hints. It's entirely possible that my question is ill-posed since I've just start studying this topic and therefore my grasp on it is not strong.

Answer 1

Thanks for the questions. Regarding questions 1 and 2, the source is K(Z,n) for the composition to be null. This is explicit on p.130, and the tetherings are not ignored. Moreover, the description of the mod 2 cohomology of Eilenberg-Mac Lane spaces appears on p.28. On p. 61, there is a discussion of "strong invariance of coliftings" which is used (often tacitly) to ignore tetherings. Question 3 gets to the core of the subject and is why I wrote the book. Clever or not, the most accessible approach I know to calculation is described in chapter 2 and used systematically in the book, including the exercises. My experience is that one has to work through a bunch of examples in order to digest this material, and it is a big help if one has a specific problem in mind.