  We study the geodesic orbit property for nilpotent Lie groups $N$ when
endowed with a pseudo-Riemannian left-invariant metric. We consider this
property with respect to different groups acting by isometries. When $N$ acts
on itself by left-translations we show that it is a geodesic orbit space if and
only if the metric is bi-invariant. Assuming $N$ is 2-step nilpotent and with
non-degenerate center we give algebraic conditions on the Lie algebra
$\mathfrak n$ of $N$ in order to verify that every geodesic is the orbit of a
one-parameter subgroup of $N\rtimes\operatorname{Auto}(N)$. In addition we
present an example of an almost g.o. space such that for null homogeneous
geodesics, the natural parameter of the orbit is not always the affine
parameter of the geodesic.
