Automorphisms of a Clifford-like parallelism

In this paper we focus on the description of the automorphism group ${\mathrm{\Gamma }}_{\parallel }$ of a Clifford-like parallelism $\parallel$ on a $3$-dimensional projective double space $(\mathbb{P}(H_F),,)$ over a quaternion skew field $H$ (with centre a field $F$ of any characteristic). We compare ${\mathrm{\Gamma }}_{\parallel }$ with the automorphism group ${\mathrm{\Gamma }}_{\mathrm{\ell }}$ of the left parallelism $$, which is strictly related to $\mathrm{A}\mathrm{u}\mathrm{t}(H)$. We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose $\parallel$ in such a way that ${\mathrm{\Gamma }}_{\parallel }$ is either properly contained in ${\mathrm{\Gamma }}_{\mathrm{\ell }}$ or coincides with ${\mathrm{\Gamma }}_{\mathrm{\ell }}$ even though $\parallel \mathrm{\ne }$.