Generalized Preparata codes and $2$-resolvable Steiner quadruple systems

We consider generalized Preparata codes with a noncommutative group operation. These codes are shown to induce new partitions of Hamming codes into cosets of these Preparata codes. The constructed partitions induce $2$-resolvable Steiner quadruple systems $S(n,4,3)$ (i.e., systems $S(n,4,3)$ that can be partitioned into disjoint Steiner systems $S(n,4,2)$). The obtained partitions of systems $S(n,4,3)$ into systems $S(n,4,2)$ are not equivalent to such partitions previously known.