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We know that there is a relation between Bockstein cohomology groups and the cohomology groups as stated in the proposition below:

Theorem(taken from Hatcher's textbook on Algebraic Topology):If $H_n(X; Z)$ is finitely generated for all $n$, then the Bockstein cohomology groups $BH^n(X; Z_p)$ are determined by the following rules:

(a) Each $Z$ summand of $H^n(X; Z)$ contributes a $Z_p$ summand to $BH^n(X; Z_p)$.

(b) Each $Z_{p^k}$ summand of $H^n(X; Z)$ with $k > 1$ contributes $Z_p$ summands to both $BH^{n−1}(X; Z_p)$ and $BH^n(X; Z_p)$.

(c) A $Z_p$ summand of $H^n(X; Z)$ gives $Z_p$ summands of $H^{n−1}(X; Z_p)$ and $H^n(X; Z_p)$ with $\beta$ an isomorphism between these two summands, hence there is no contribution to $BH^∗(X; Z_p)$.

So is there a similar statement for Homology groups?

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    That result is proved quite in detail from the UCT for cohomology in the book. Have you tried following the argument with homology to see where it breaks? – 2017-02-13
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    @MarianoSuárez-Álvarez I have not really looked at the proof of this statement. The reason I asked this question was because I came across such a theorem while calculating the higher homotopy groups of spheres using serre fibration. So I am interested in just knowing the statement at present(because this is more like a tangential result to what I am reading at present). If this is not in spirit of this site I am truly sorry and I will delete the question. – 2017-02-13

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