Automorphisms of a Clifford-like parallelism

In this paper we focus on the description of the automorphism group Γ of a Clifford-like parallelism on a 3-dimensional projective double space over a quaternion skew field H (with centre a field F 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 the Clifford-like parallelism in such a way that Γ is either properly contained in Γ ' or coincides with Γ ' even though the corresponding parallelisms are different.