I wonder if the group cohomology of a finite group $G$ with coefficients in $\mathbb{Z}$ is finite. This statement may be too strong. I am interested in, for instance, dihedral group. $ G=D_{2n}=\langle a,b | \ a^n=b^2=abab=e \rangle $ Assume that $a$ acts trivially and $b$ acts as $-id$ on $\mathbb{Z}$.
First cohomology is $\mathbb{Z}^G=0$. The second cohomology already seems quite involved to me.
I read several post about group cohomology on StackExchange and MathOverflow, but I still have trouble computing explicit example and getting intuition behind the concept.
Thank you for your assistance.