  The complete classification of the orbits on subspaces under the action of
the projective stabiliser of (classical) algebraic varieties is a challenging
task, and few classifications are complete. We focus on a particular action of
$\PGL(2,q^2)$ (and $\PSL(2,q^2)$) arising from the Hermitian Veronese curve in
$\PG(3, q^2)$, a maximal rational curve embedded on a smooth Hermitian surface
with some fascinating properties. The study of its orbits leads to a new
construction of quasi-Hermitian surfaces: sets of points with the same
combinatorial and geometric properties as a non-degenerate Hermitian surface.
