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We construct finite-dimensional Hopf algebras whose coradical is the groupalgebra of a central extension of an abelian group. They fall into familiesassociated to a semisimple Lie algebra together with a Dynkin diagramautomorphism. We show conversely that every finite-dimensional pointed Hopfalgebra over a non-abelian group with non-simple infinitesimal braiding of rankat least 4 is of this form.  We follow the steps of the Lifting Method by Andruskiewitsch--Schneider. Ourstarting point is the classification of finite-dimensional Nichols algebrasover non-abelian groups by Heckenberger--Vendramin, which consist of low rankexceptions and large rank families. We prove that the large rank families arecocycle twists of Nichols algebras constructed by the second author as foldingsof Nichols algebras of Cartan type over abelian groups by outer automorphisms.This enables us to give uniform Lie-theoretic descriptions of the large rankfamilies, prove generation in degree one and construct liftings. We also showthat every lifting is a cocycle deformation of the corresponding coradicallygraded Hopf algebra using an explicit presentation by generators and relationsof the Nichols algebra.  On the level of tensor categories, we construct families of graded extensionsof the representation category of a quantum group by a group of diagramautomorphism.

Pointed Hopf algebras over non-abelian groups with non-simple standard braidings