Coindex and Rigidity of Einstein Metrics on Homogeneous Gray Manifolds

Any 6-dimensional strict nearly Kahler manifold is Einstein with positive scalar curvature. We compute the coindex of the metric with respect to the Einstein-Hilbert functional on each of the compact homogeneous examples. Moreover, we show that the infinitesimal Einstein deformations on F-1,F-2 = SU(3)/T-2 are not integrable into a curve of Einstein metrics.