Existence of integral Hopf orders in twists of group algebras

We find a group-theoretical condition under which a twist of a group algebra, in Movshev's way, admits an integral Hopf order. Let K be a (large enough) number field with ring of integers R. Let G be a finite group and M an abelian subgroup of G of central type. Consider the twist J for K\hspace{-0.8pt}G afforded by a non-degenerate 2-cocycle on the character group \widehat{M}. We show that if there is a Lagrangian decomposition \widehat{M} \simeq L \times \widehat{L} such that L is contained in a normal abelian subgroup N of G, then the twisted group algebra (K\hspace{-0.8pt}G)_J admits a Hopf order X over R. The Hopf order X is constructed as the R-submodule generated by the primitive idempotents of K\hspace{-1.1pt}N and the elements of G. It is indeed a Hopf order of K\hspace{-0.8pt}G such that J^{\pm 1} \in X \otimes_R X. Furthermore, we give some criteria for this Hopf order to be unique. We illustrate this construction with several families of examples. As an application, we provide a further example of simple and semisimple complex Hopf algebra that does not admit integral Hopf orders.