Automorphisms of a $({24}_4,{32}_3)$-configuration on the Schur quartic

We answer the automorphism question raised by Naskręcki and Pokora for their $({24}_4,{32}_3)$-configuration on the Schur quartic. The stabilizer of either $24$-line half is isomorphic to $W(D_4)⋊C_3$, where $C_3$ acts by even triality; the full projective automorphism group has order $1152$. The $D_4$ model also gives an intrinsic coloring and shortens the incidence check.