Einstein with skew-torsion connections on Berger spheres

The invariant metric affine connections on Berger spheres which are Einstein with skew-torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. Notably, every Berger sphere with Lorentzian signature admits invariant metric affine connections which are Einstein with skew-torsion except for dimension 3. For Riemannian signature, the existence of such connections strongly depends on the dimension of the sphere and on the scale of the deformation used for the Berger metric. In particular, there are Riemannian Berger spheres, not Einstein, which admit invariant Einstein with skew-torsion affine connections.