  Motivated by the appearance of penumbral moonshine, and by evidence that
penumbral moonshine enjoys an extensive relationship to generalized monstrous
moonshine via infinite products, we establish a general construction in this
work which uses singular theta lifts and a concrete construction at the level
of modules for a finite group to translate between moonshine in weight one-half
and moonshine in weight zero. This construction serves as a foundation for a
companion paper in which we explore the connection between penumbral Thompson
moonshine and a special case of generalized monstrous moonshine in detail.
