Module categories over equivariantized tensor categories

For a finite tensor category $\mathcal C$ and a Hopf monad $T\colon\mathcal C\to\mathcal C$ satisfying certain conditions we describe exact indecomposable left $\mathcal C^T$-module categories in terms of left $\mathcal C$-module categories and some extra data. We also give a $2$-categorical interpretation of the process of equivariantization of module categories.