Frobenius extensions of subalgebras of Hopf algebras

We consider when extensions S subset of R of subalgebras of a Hopf algebra are beta-Frobenius, that is Frobenius of the second kind. Given a Hopf algebra H, we show that when S subset of R are Hopf algebras in the Yetter-Drinfeld category for H, the extension is beta-Frobenius provided R is finite over S and the extension of biproducts S * H subset of R * H is cleft. More generally we give conditions for an extension to be beta-Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants. We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.