Let be an equivariant embedding of a connected reductive group over an algebraically closed field of positive characteristic. Let denote a Borel subgroup of . A -Schubert variety in is a subvariety of the form , where is a -orbit closure in . In the case where is the wonderful compactification of a group of adjoint type, the -Schubert varieties are the closures of Lusztig's -stable pieces. We prove that admits a Frobenius splitting that compatibly splits all the -Schubert varieties. Moreover, any -Schubert variety admits stable Frobenius splittings along ample divisors in case X is projective. Although this indicates that -Schubert varieties have nice singularities we give an example, in the wonderful compactification of a group of adjoint type, which is not normal. Finally we also extend the Frobenius splitting results to the more general class of -Schubert varieties.