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 $d_1$.

A general algebraic approach to Steenrod operations(whose title I may be imperfectly remembering), where the relevant operation is called $\beta_2$. $\endgroup$