In this paper we focus on the description of the automorphism group $\Gamma_{ \parallel}$ of a Clifford-like parallelism $\parallel$ on a $3$-dimensional projective double space $\bigl(\mathbb{ P}(H_F),{ \mathrel{ \parallel_{ \ell}}},{ \mathrel{ \parallel_{ r}}}\bigr)$ over a quaternion skew field $H$ (with centre a field $F$ of any characteristic). We compare $\Gamma_{ \parallel}$ with the automorphism group $\Gamma_{ \ell}$ of the left parallelism $\mathrel{ \parallel_{ \ell}}$, which is strictly related to $\mathrm{ Aut}(H)$. 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 $\Gamma_{ \parallel}$ is either properly contained in $\Gamma_{ \ell}$ or coincides with $\Gamma_{ \ell}$ even though ${ \parallel} \neq{ \mathrel{ \parallel_{ \ell}}}$.
Categories: math.AG