Examples of Tate cohomology rings  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 ?   
 A: The ring structure on mod $p$ Tate cohomology can be split up into 4 parts:


*

*Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n,m\geq0$.  These just come from products on ordinary cohomology.

*Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n\geq0$, $m<0$, and $n+m<0$.  These are given by cap products when we identify $\hat{H}^m$ with the dual of $H^{-m-1}$.

*Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n\geq0$, $m<0$, and $n+m\geq 0$.

*Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n,m<0$.


In fact, for most groups, all products of type 3 and 4 vanish.  This is true, for instance, if the center of a $p$-Sylow subgroup of $G$ has rank greater than 1: see this paper of Benson and Carlson (they also show that all products of type 3 vanish iff all products of type 4 vanish).  By Chouinard's theorem, this implies that for arbitrary non-periodic groups, all elements in negative mod $p$ Tate cohomology are nilpotent.
A: Bellezza's thesis is here
http://www.abdn.ac.uk/~mth192/html/archive/bellezza.html
A: 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Also, there is an unpublished/unfinished paper of Weiss, found here: http://www.math.uwo.ca/~schebolu/research/Jan/tateprop.pdf  , which computes $\hat{H}^*(\mathbb{Z}_2^r,\mathbb{Z}_2)$.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 $|z|=4$.
A: Artin and Tate showed in their class field theory book that a finite group has periodic mod $p$ cohomology if and only if its Sylow $p$–group is either a cyclic group or a generalised quaternion group. Swan proved later that a group has periodic cohomology for all primes if and only if it acts freely on a finite CW-complex which has the homotopy type of a sphere.
