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The Bockstein SS is obtained from the exact sequence $$0\to\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to\mathbb{Z}/2\to 0$$ with $E_1^p=H^p(X,\mathbb{Z}/2)$ and the differential $d_1=Sq^1$.

How to identify the differentials $d_2$ for the $E_2$-page without knowing $H^*(X,\mathbb{Z})$ in advance?

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    $\begingroup$ $d_2$ is described by a “secondary” cohomology operation. You can read quite a bit about it in May’s A general algebraic approach to Steenrod operations (whose title I may be imperfectly remembering), where the relevant operation is called $\beta_2$. $\endgroup$
    – Eric Peterson
    Commented Jun 24, 2022 at 4:16

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The $E_1$ page does not tell you what the higher differentials will be, and you will have to know at least something about the integral cohomology. Consider the case when $X$ is a Moore space $M(1,\mathbb{Z}/2^n)$ which you may prefer to think of as a circle with a disc attached via a $\times 2^n$ map. In each case the $E_1$-page is the same, independent of $n$. If $n=1$, the differential $d_1$ is non-zero. If $n>1$ then the differential $d_1=Sq^1$ is zero. The only differential that is non-zero in this case is $d^n$. The cases $n=2,3,4,\ldots$ cannot be distinguished by knowing the $E_1$-page and $d_1$.

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  • $\begingroup$ Is the $d_2$ some kind of cohomological operation for which some references could be found? $\endgroup$
    – Nanjun Yang
    Commented Jun 24, 2022 at 1:40
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    $\begingroup$ In general, higher differentials are higher cohomology operations, defined only on the kernels of the differentials below them. If $a$ is a mod-2 cohomology class with $Sq^1(a)=0$, this says that the coboundary $b$ of an integral lift of $a$ is divisible by~2. The secondary operation in this case looks at $b/2$ as a mod-2 cocycle; this is defined only because $b$ was 0 mod-2. $\endgroup$
    – IJL
    Commented Jun 24, 2022 at 14:54

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