Let S be a commutative local ring of characteristic p, which is not a field, S* the multiplicative group of S, W a subgroup of S*, G a finite p-group, and $S^{\lambda }G$ a twisted group ring of the group G and of the ring S with a 2-cocycle λ ∈ Z²(G,S*). Denote by $In{d}_m\left(S^{\lambda }G\right)$ the set of isomorphism classes of indecomposable $S^{\lambda }G$-modules of S-rank m. We exhibit rings $S^{\lambda }G$ for which there exists a function $f_{\lambda }:\mathbb{N}\to \mathbb{N}$ such that $f_{\lambda }\left(n\right)\ge n$ and $In{d}_{f_{\lambda }\left(n\right)}\left(S^{\lambda }G\right)$ is an infinite set for every natural n > 1. In special cases $f_{\lambda }\left(\mathbb{N}\right)$ contains every natural number m >...