Abelian quotients and the largest orbit sizes of linear groups of odd order

Let $G$ be a finite nonabelian group of odd order, let $p$ be the smallest prime divisor of $\mathrm{\mid }G\mathrm{\mid }$, and let $V$ be a finite faithful completely reducible $G$-module, possibly of mixed characteristic. Suppose that $M$ is the largest orbit size in the action of $G$ on $V$. It is known that $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }\le M$, and that equality holds only for abelian groups if $\mathrm{\mid }G\mathrm{\mid }$ is odd. We prove that $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }\le M\mathrm{/}p$, unless $p=3$ and $G$ is the direct product of an abelian group and a group isomorphic to $\mathrm{\Gamma }(2^3)$ or to $\mathrm{\Gamma }(2^3)\times C_3$ acting on $V$ in a specific way, in which case $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }=3M\mathrm{/}7$. In particular, $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }\le 3M\mathrm{/}7$ for every nonabelian group of odd order. If $\mathrm{\mid }V\mathrm{\mid }$ is odd, then $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }=M\mathrm{/}p$ holds if and only if $\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }=p$; more precisely, nilpotent groups then have regular orbits, whereas non-nilpotent groups satisfy $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }\le 3M\mathrm{/}13$, and even $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }<M\mathrm{/}(2p)$ if $p\ge 5$. The bound $3M\mathrm{/}13$ is best possible.