  Let k be a field and let G be a finite group. By a theorem of D.Benson,
H.Krause and S.Schwede, there is a canonical element in the Hochschild
cohomology of the Tate cohomology HH^{3,-1} H*G with the following property:
Given any graded H*G-module X, the image of the canonical element in
Ext^{3,-1}(X,X) is zero if and only if X is isomorphic to a direct summand of
H*(G,M) for some kG-module M.
  We investigate this canonical element in certain special cases, namely that
of (finite) abelian p-groups and the quaternion group. In case of
non-triviality of the canonical element, we also give examples of
non-realizable modules X.
