I am looking at the computation of $\pi_5(S^3) = \mathbb{Z}/2\mathbb{Z}$ and $\pi_6(S^3) = \mathbb{Z}/12\mathbb{Z}$ in Hatcher's spectral sequence notes and I am stuck trying to get off the ground with it. The question below does not actually get into the nitty gritty of the spectral sequence. Also, I apologies but I am a total noobie with this stuff so I am sure my problem is not so interesting but I appreciate the help orienting myself. I imagine that my problem is that I have not read much of the rest of the chapter/book in so if anyone has read through these notes and wants to point me to an earlier part that clears up my confusion, that would be highly appreciated.
We begin by looking at the fibration in the Postnikov tower $K(\pi_4(S^3),4) \to X_4 \to X_3 = K(\pi_3(S^3),3)$, and we have already determined that $\pi_4(S^3) = \mathbb{Z}/2\mathbb{Z}$. Hatcher says that the game plan is to determine everything about the Serre spectral sequence for this fibration for $p+q \leq 8$ and partial information for $p+q=9$. We will need to look at cohomology with both $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}$ coefficients, but when using $\mathbb{Z}$ coefficients we only need to compute the groups modulo odd torsion.
We already know that $H^*(K(\mathbb{Z},3); \mathbb{Z}/2\mathbb{Z})$ is a polynomial algebra with generators $\iota_3, Sq^2(\iota_3), Sq^4Sq^2(\iota_3)...$ I am confused about how Hatcher uses the Bockstein homomorphism $\beta = Sq^1$ to figure out what $H^*(K(\mathbb{Z},3);\mathbb{Z})$ is (modulo odd torsion) in these dimensions as well as how determining the coefficient change map $H^*(K(\mathbb{Z}, 3);\mathbb{Z}) \to H^*(K(\mathbb{Z}, 3);\mathbb{Z}/2\mathbb{Z})$?
Additionally, I am confused as to how Hatcher is computing the Bockstein homomorphism - specifically, how he knows that: $$ Sq^3 \iota_3 = \iota_3^2 $$ or $$ Sq^5Sq^2 \iota_3 = (Sq^2 \iota)^2 $$
One point of confusion is that $\beta : H^n(K(\mathbb{Z}, 3);\mathbb{Z}/2\mathbb{Z}) \to H^{n+1}(K(\mathbb{Z}, 3);\mathbb{Z}/2\mathbb{Z})$, so I don't see the $\mathbb{Z}$-coefficient group appearing.