Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout–Metz unitals

It is shown that for every semifield spread in PG(3,q) and for every parabolic Buekenhout–Metz unital, there is a collineation group of the associated translation plane that acts transitively and regularly on the affine points of the parabolic unital. Conversely, any spread admitting such a group is shown to be a semifield spread. For hyperbolic Buekenhout unitals, various collineation groups of translation planes admitting such unitals and the associated planes are determined.