Automorphisms of a Clifford-like parallelism

We focus on the description of the automorphism group Γ∥ of a Clifford-like parallelism ∥ on a 3-dimensional projective double space ((HF), ∥∥, ∥r) over a quaternion skew field H (of any characteristic). We compare Γ∥ with the automorphism group Γ∥ of the left parallelism ∥∥, which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose in such a way that Γ∥ is either properly contained in Γ∥ or coincides with Γ∥ even though ∥.