I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite.

Let us assume we are given a group $G$ and a $\mathbb{Z}G$-module $A$. Then we can define the ** cohomology groups** of $G$ as $$H^n(G,A) = \mathop{Ext}\nolimits^n_{\mathbb{Z}G}(\mathbb{Z}, A).$$

I am interested in techniques for calculating or bounding the cardinality of the cohomology groups *in case they are finite*.

One simple example can be given for cyclic groups. If $G$ is cyclic of order $k$ and $A$ is a trivial $\mathbb{Z}G$-module (i.e., $gm = m$ for all $g \in G$ and $m \in \mathbb{Z}$), then $H^0(G,A) = A$, $H^{2n-1}(G,A) = A[k]$, and $H^{2n}(G,A) = A/kA$, where $n \geq 1$. So this means, in particular, that $|H^{2n}(G,\mathbb{Z})| = k$. But the above theorem gives much more then what I ask for, namely the structure of $H^n(G,A)$. This is of course very well as long as I can get the cardinality from it.

So my question is: *What techniques are known for calculating or bounding the cardinality of $H^n(G,A)$ in case it is a finite group?*