Automorphism groups of pre-Lie Witt doubles

Let ${\mathscr A}$ be a finite-dimensional associative commutative $d$-simple algebra but not a field with nonzero derivation $d$ over an algebraically closed field $F$. We describe the automorphisms of the left-symmetric Witt doubles ${\mathscr A}_d$ and ${\mathscr W}_d({\mathscr A})$ over $F$ modulo the automorphisms of ${\mathscr A}$ almost commuting with $d$.