  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. In particular, if the canonical element vanishes,
then every module is a direct summand of a realizable H*(G)-module.
  We prove that the converse of that last statement is not true by studying in
detail the case of generalized quaternion groups. Suppose that k is a field of
characteristic 2 and G is generalized quaternion of order 2^n with n>2. We show
that the canonical element is non-trivial for all n, but there is an
H*(G)-module detecting this non-triviality if and only if n=3.
