Invariant geodesic orbit Finsler (α,β) metrics F which arise
from Riemannian geodesic orbit metrics α on spheres are determined. The
relation of Riemannian geodesic graphs with Finslerian geodesic graphs proved
in a previous work is now illustrated with explicit constructions. Interesting
examples are found such that (G/H,α) is Riemannian geodesic orbit space,
but for the geodesic orbit property of (G/H,F) the isometry group has to be
extended. It is also shown that projective spaces other than RPn
do not admit invariant purely Finsler (α,β) metrics