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?

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?