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.

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.