Weakly symmetric pseudo–Riemannian nilmanifolds

In an earlier paper we developed the classification of weakly symmetric pseudo–Riemannian manifolds $G/H$, where $G$ is a semisimple Lie group and $H$ reductive subgroup. We derived from cases compact. As consequence obtained Lorentz signature $(n-1,1)$ trans-Lorentzian $(n-2,2)$. Here work out pseudo-Riemannian nilmanifolds $G/H$ for case $G=N \rtimes H$ with compact $N$ nilpotent. It turns that there plethora new examples merit further study. Starting Riemannian case, see just when given involutive automorphism extends to $G$, show any two such extensions result in isometric nilmanifolds. The results are tabulated last sections paper.