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We study certain subgroups of the full group of Hopf algebra automorphisms of twisted tensor biproducts.

We study certain subgroups of the full group of monoidal Hom-Hopf algebra automorphisms of a Hom-biproduct, which gives a Hom-version of Radford’s results.

We investigate how the category of Hom-entwined modules can be made into a monoidal category. The sufficient and necessary conditions making the category of Hom-entwined modules have a braiding are given. Also, we formulate the concept of Hom-cleft extension for a Hom-entwining structure, and prove that if $(A, \alpha)$ is a $(C,\gamma)$-cleft extension, then there is an isomorphism of Hom-algebras between $(A, \alpha)$ and a crossed product Hom-algebra of $A^{coC}$ and $C$.

Using the theory of group corings, we study (graded) Morita contextsassociated to a comodule over a group coring, which generalize andunify some classical morita contexts. Some applications of our theoryare also discussed.