Frobenius and Mashke type Theorems for Doi-Hopf modules and entwined   modules revisited: a unified approach
  We study when induction functors (and their adjoints) between categories of Doi-Hopf modules and, more generally, entwined modules are separable, resp. Frobenius. We present a unified approach, leading to new proofs of old results by the authors, as well as to some new ones. Also our methods provide a categorical explanation for the relationship between separability and Frobenius properties. 
Tomasz Brzezinski, S. Caenepeel, G. Militaru and Shenglin Zhu