A coset partition of a group G is a set partition of G into finitely many left cosets of one or more subgroups. A driving force in this research area is the Herzog–Schönheim Conjecture [15], which states that any nontrivial coset partition of a group contains at least two cosets with the same index. Although many families of groups have been shown to satisfy the conjecture, it remains open.

A Steiner coset partition of G, with respect to distinct subgroups $H_1,\dots ,H_r$, is a coset partition of G that contains exactly one coset of each $H_i$. In the quest of a more structural version of the Herzog–Schönheim Conjecture, it was shown that there is no Steiner coset partition of G with respect to any $r\geq 2$ subgroups $H_i$ that mutually commute [1]. In this article, we show that this result holds for $r=4$ mutually commuting subgroups provided that G does not have $C_2\times C_2\times C_2$ as a quotient, where $C_2$ is the cyclic group of order $2$. We further give an explicit construction of Steiner coset partitions of the n-fold direct product $G^*=C_p\times \ldots \times C_p$ for p prime and $n\geq 3$. This construction lifts to every group extension of $G^*$.