salient features of this example. Let $\nu$ be an automorphism of $G$ with $\nu^2=1$. Then $G$ acts on the set of *generalized involutions* $$\cI_{G,\nu} {\stackrel{\rm def}{=}}\{ \omega \in G : \omega^{-1} = \nu(\omega)\}$$ by the twisted conjugation $g: \omega \mapsto g\cdot \omega \cdot \nu(g)^{-1}$. We write $$C_{G,\nu}(\omega) {\stackrel{\rm def}{=}}\{ g\in G : g\cdot \omega \cdot \nu(g)^{-1} = \omega \}$$ to denote the stabilizer of $\omega \in \cI_{G,\nu}$ under this action, and say that a model $\{ \lambda_i : H_i \to {\mathbb{C}}\}$ is a *generalized involution model* (or *GIM* for short) with respect to $\nu$ if each $H_i$ is the stabilizer $C_{G,\nu}(\omega)$ of a generalized involution $\omega \in \cI_{G,\nu}$, with each twisted conjugacy class in $\cI_{G,\nu}$ contributing exactly one subgroup. The model in the example is a GIM with respect to the inverse transpose automorphism of $G(r,n)$.

In [@Ma1; @Ma], the second author classified which finite complex reflection groups have GIMs. Subsequently, the first author discovered an interesting reformulation of this classification, which suggests that these results are most naturally interpreted in the broader context of *projective reflection groups*. These groups were introduced in [@Ca1] and studied, for example, in [@BC3]. They include as an important special case an infinite series 