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A particular paper that comes to mind is:
On orders and vanishing of integral cohomology groups, Angelina Chin
Here she obtains a recurrence relation involving the orders of the cohomology groups of a finite group $G$ whose quotient by a normal subgroup is cyclic.

1. Using Hopf's formula or $H^2(G,\mathbb{Z})\cong Hom(G,\mathbb{Q}/\mathbb{Z})$ should get you the orders of groups for 2nd cohomology.
2. You can check to see if your group has periodic cohomology (period $d$), and then $|H^n(G,\mathbb{Z})|=|H^{n+d}(G,\mathbb{Z})|=|G|$ for some $n$.
3. (Trivially) You can check to see if your group is cohomologically trivial, and then its cohomology vanishes.
4. For abelian groups you can probably get some information from the explicit calculation of $H_*(G,\mathbb{Z}_p)$, given in Theorem V.6.6 of Ken Brown's Cohomology of Groups textbook.