You might find some useful information in the Lausanne thesis of Alain Clément:
[http://doc.rero.ch/record/482/files/Clement_these.pdf][1]
In particular, he gives an account of Cartan's results in Chapter 2, then describes a C++ program for computing integral (co)homology of certain ($2$-local) Eilenberg-Mac Lane spaces in Chapter 3. An appendix lists the integral (co)homology groups of $K(\mathbb{Z}_2,2)$, $K(\mathbb{Z}_2,3)$, $K(\mathbb{Z}_4,2)$ and $K(\mathbb{Z}_4,3)$ up to degree $200$. [1]: http://doc.rero.ch/record/482/files/Clement_these.pdf