If $G$ is a finite group then there is the so-called Bockstein spectral sequence $$E_2^n = H^n(G,\mathbb{F}_p) \Rightarrow \begin{cases} \mathbb{F}_p & n =0 \newline 0 & n>0\end{cases}$$ that can be used to compute the integral cohomology out of the mod-$p$ cohomology. In more detail, each non-zero element of $d_r(E_r^{n-1})\subseteq E_r^n$ corresponds to a direct $\mathbb{Z}/p^r$-summand of $H^n(G,\mathbb{Z})$ (Corollary 5.9.12 in Weibel's Homological Algebra book).

Now my question is whether it is possible to compute the integral cohomology ring of $G$ if the mod-$p$ cohomology rings for the primes dividing $|G|$ and the differentials in the associated Bockstein spectral sequences are known ?