Modules with finitely generated cohomology

Let G be a finite group and k a field of characteristic p. It is conjectured in a paper of the first author and John Greenlees that the thick subcategory of the stable module category StMod(kG) consisting of modules whose cohomology is finitely generated over H⁎(G,k) is generated by finite dimensional modules and modules with no cohomology. If the centraliser of every element of order p in G is p-nilpotent, this statement follows from previous work. Our purpose here is to prove this conjecture in two cases with non p-nilpotent centralisers. The groups involved are Z/3r×Σ3 (r⩾1) in characteristic three and Z/2×A4 in characteristic two. As a consequence, in these cases the bounded derived category of C⁎BG (cochains on BG with coefficients in k) is generated by C⁎BS, where S is a Sylow p-subgroup of G.