Collineation groups of the intersection of two classical unitals

Kestenband proved in \cite{K1} that there are only seven pairwise
non-isomorphic Hermitian intersections in
the desarguesian projective plane $\PG(2,\qq)$ of square order $\qq$.
His classification is based on the study of the minimal polynomials
of the matrices associated with the curves and
leads to results of purely combinatorial nature: in fact, two
Hermitian intersections from the same class might not be projectively
equivalent in $\PG(2,\qq)$ and might have different
collineation groups. The projective classification of Hermitian
intersections in $\PG(2,\qq)$ is the main goal in this paper.
It turns out that each of Kestenband's classes consists of projectively
equivalent Hermitian intersections.
A complete classification of the linear collineation groups
preserving a Hermitian intersection is also given.