Every book on modular representation theory of finite groups introduces p-modular systems and describes how to reduce an ordinary representation $U$ to obtain one in characteristic p (call it $\overline{U}$), by choosing a lattice. They all prove that the resulting $kG$- module is not unique, but is well defined in the Grothendieck group.

Are there any general results about how rigid this process is. For example for which $U$ is it the case that $\overline{U}$ is uniquely determined up to isomorphism? Can any composition factor of $\overline{U}$ be forced to the socle by choosing an appropriate lattice? Can the number of indecomposable summands of $\overline{U}$ change depending on the choice of lattice? I haven't been able to find any results like these.