Let $G$ be a finite group generated by (pseudo-) reflections in a complex vector space and let $g$ be any linear transformation which normalises $G$. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset $gG$, a subquotient of $G$ which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in $G$ of certain elements of the coset. A criterion is also given in terms of the invariant degrees of $G$ for an integer to be regular for $G$. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.