Let $\mathcal{H} \subset \mathbb{P}^2(\mathbb{F}_{q^2})$ be the Hermitian curve, defined (up to projective equivalence) by $$ X^{q+1} + Y^{q+1} + Z^{q+1} = 0. $$ Its automorphism group is $\mathrm{PGU}(3,q)$, a subgroup of $\mathrm{PGL}(3,q^2)$.
Nondegenerate conics in $\mathbb{P}^2(\mathbb{F}_{q^2})$ correspond to nonsingular quadratic forms, and $\mathrm{PGL}(3,q^2)$ acts transitively on them. However, under the smaller group $\mathrm{PGU}(3,q)$, they split into several orbits.
Is it possible to classify these $\mathrm{PGU}(3,q)$-orbits and relate each orbit to the number of $\mathbb{F}_{q^2}$-rational intersection points with $\mathcal{H}$? In particular:
Do certain orbits force the intersection size $|\mathcal{C} \cap \mathcal{H}|$ to be a fixed value, such as $q+1$?
Has this ``relative orbit'' problem—i.e., orbit decomposition under a subgroup and its geometric consequences—been systematically studied in the literature?
I am especially interested in whether the intersection size is constant on each $\mathrm{PGU}(3,q)$-orbit of conics.
