Nonequivalent degree-k induced faithful representations φk: Sn ↪ Rk(Zn), existence and classification

Identify the complete set of nonequivalent degree-k induced faithful Riordan representations φk: Sn ↪ Rk(Zn) for integers n ≥ 4 and k ≤ n, if such representations exist; each φk is obtained by truncating a faithful Riordan array representation over Zn to degree k via πk: R(Zn) → Rk(Zn).

Background

The authors classify degree-2 Riordan representations of S3 over Z3, finding exactly two nonequivalent representations. Extending this to larger symmetric groups and higher truncation degrees is a natural but nontrivial generalization.

This question requests both existence and classification of nonequivalent induced faithful representations into truncated Riordan groups Rk(Zn) when n ≥ 4, building on the methodology developed in Section 3.