The cohomological decomposition conjecture for crystallographic groups with cyclic holonomy

Let $G$ be a finite cyclic group and $L$ a finitely generated $\mathbb{Z}G$-lattice. For the split crystallographic group $L\rtimes G$, consider its integral cohomology and the cohomology of $G$ with coefficients in the cohomology of $L$. Cohomological decomposition conjecture. For every $k\geq 0$, there should be an isomorphism

$$\mathrm{H}^k(L\rtimes G,\mathbb{Z})\cong\bigoplus_{i+j=k}\mathrm{H}^i(G,\mathrm{H}^j(L,\mathbb{Z})).$$

The conjecture extends the known prime-order cyclic holonomy case. It is refuted by counterexamples, including examples in dimensions at most $5$ and an example with odd-order holonomy $G\cong\mathbb{Z}_9$ in dimension $8$.