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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 $$H^n(G,A) = \mathop{Ext}\nolimits^n_{\mathbb{Z}G}(\mathbb{Z}, A).$$
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?