Crossed module on 3-Hom–Lie–Rinehart superalgebras

This work aims to introduce the concept of [Formula: see text]-Hom–Lie–Rinehart superalgebras, a twisted version of [Formula: see text]-Lie–Rinehart superalgebras, and to systematically explore their representations. Additionally, we investigate the connections between a Hom–Lie–Rinehart superalgebra and its associated [Formula: see text]-Hom–Lie–Rinehart superalgebra. Finally, we define crossed modules of [Formula: see text]-Hom–Lie–Rinehart superalgebras and provide their characterizations.