a GIM if and only if $\gcd(p,n) = 1$.

Combining Theorems \[thm2\] and \[thm3\] shows that to completely determine which projective reflection groups $G(r,p,q,n)$ have GIMs, it remains only to consider groups of the form $$G(2r,1,2q,2n)\quad\text{or}\quad G(2r,2,2q,2n)\quad\text{or}\quad G(8r+4,4,8q+4,8n+4).$$ (Of course we only need to consider the first two types when $2n>2$). We state some conjectures concerning which of these groups should have GIMs at the end of Section \[last-sect\]. This research continues a line of inquiry taken up by a number of people in the past few decades. Researchers originally considered *involution models*, which are simply GIMs defined with respect to the identity automorphism. Inglis, Richardson, and Saxl described an elegant involution model for the symmetric group in [@IRS] (which is precisely the model in Example \[exa2\] when $r=1$). In his doctoral thesis, Baddeley [@B91] classified which finite Weyl groups have involution models. Vinroot [@V] extended this classification to show that the finite Coxeter groups with involution models are precisely those of type $A_n$, $BC_n$, $D_{2n+1}$, $F_4$, $H_3$, and $I_2(m)$. In extending this classification to reflection groups, it is natural to consider generalized involution models, since only groups whose representations are all realizable over the real numbers can possess involution 