Let \({\mathcal {F}} \, = \, (\dots {\mathop {\rightarrow }\limits ^{\partial _{n+1}}} {\mathcal {F}}_n {\mathop {\rightarrow }\limits ^{\partial _n}} {\mathcal {F}}_{n-1}{\mathop {\rightarrow }\limits ^{\partial _{n-1}}} \dots \dots {\mathop {\rightarrow }\limits ^{\partial _1}} {\mathcal {F}}_0 \rightarrow {\mathfrak {R}} \rightarrow 0)\) be a free resolution over the group ring \({\mathfrak {R}}[\Phi ]\) where \({\mathfrak {R}}\) is commutative and \(\Phi \) is finite. The \(n^{th}\) syzygy \(\Omega _n^{{\mathfrak {R}}[\Phi ]}\) is the stable class of \(\textrm{Im}(\partial _n)\) and has a tree structure with roots which do not extend infinitely downwards. We show that \(\Omega _3^{{\mathfrak {R}}[Q_{8p}]}\) has infinitely many isomorphically distinct modules at the minimal level when \(\,{\mathfrak {R}} = {\mathbb {Z}}[C_\infty ]\) is the integral group ring of the infinite cyclic group and \(Q_{8p}\) is the quaternion group of order 8p where \(p \ge 3\) is prime. This poses severe difficulties in attempting to solve the D(2) problem of CTC Wall for the groups \(C_\infty \times Q_{8p}\)
