Finite Non-$p$-Groups with 5 Automorphism Orbits

We obtain a characterization of finite non-$p$-groups with five automorphism orbits. In the solvable case of Fitting height $3$, the groups are precisely $S_4$ and $C_2^4\rtimes D_{10}$. Moreover, the corresponding fixed-point subspaces form a spread in a finite projective space. For Fitting height