Realizability of modules over Tate cohomology

Let k k be a field and let G G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology γ G ∈ H H 3 , − 1 H ^ ∗ ( G , k ) γ _G∈ H\!H3,-1 H^*(G,k) with the following property. Given a graded H ^ ∗ ( G , k ) H^*(G,k) -module X X , the image of γ G γ _G in Ext H ^ ∗ ( G , k ) 3 , − 1 ⁡ ( X , X ) Ext3,-1H^*(G,k)(X,X) vanishes if and only if X X is isomorphic to a direct summand of