  In this paper we continue the study of the nonvanishing minors property (NVM)
initiated by Garcia, Karaali and Katz, for the compressed Fourier matrix
attached to a subgroup $H$ of the multiplicative group of a finite field
$\mathbb{F}_q$ and a character $\chi$ defined over $H$. Here we provide a
characterization of this aforementioned property for \textit{symmetries}
arising from an index-3 subgroup $H$ and a nontrivial character $\chi$.
