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?



Higher order differentials of Bockstein spectral sequence
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 $d\_1$.


