If $G$ is a finite group with periodic cohomology then the Tate cohomology ring can be easily computed to be the localization $\hat{H}^\ast(G,\mathbb{Z}) = H^\ast(G,\mathbb{Z})\_{(z)}$ where $z$ is a unit of minimal positive degree. Examples are 


* Cyclic group: 


$\hat{H}^\ast(C\_n,\mathbb{Z}) = \mathbb{Z}[z,z^{-1}]/(nz), |z|=2$
* Quaternion group:


$\hat{H}^\ast(Q\_{2^n},\mathbb{Z}) = \mathbb{Z}[z,z^{-1},a,b]/(2^nz, 2a,2b), |z| =4, |a| = |b| =2$
* Binary icosahedral group: 


$\hat{H}^\ast(I,\mathbb{Z}) = \mathbb{Z}[z,z^{-1}]/(120z), |z|=4$


The same formular holds in mod-$p$ cohomology, if the cohomology of $G$ is $p$-periodic. 


*Question:* Are there computations of integral or mod-$p$ Tate cohomology rings of finite groups with non-periodic cohomology in the literature ? 



Examples of Tate cohomology rings 
Antonio Bellezza's PhD thesis (Pisa, 2002) computes the ring structure of the Tate cohomology for $\mathbb{Z}/p^a\times\mathbb{Z}/p^b$, and also the mod-p Tate cohomology of $\mathbb{Z}\_p^2$. The title is *Integral Duality and the Structure of Tate Cohomology Rings*.


Also, there is an unpublished/unfinished paper of Weiss, found here: <http://www.math.uwo.ca/~schebolu/research/Jan/tateprop.pdf> , which computes $\hat{H}^\*(\mathbb{Z}\_2^r,\mathbb{Z}\_2)$.


And just to add to your current list (periodic groups): $\hat{H}^\*(S\_3,\mathbb{Z}\_3)=\Lambda[x]\otimes\mathbb{Z}\_3[z,z^{-1}]$ with $|x|=3$ and $|z|=4$.


