  We investigate the generalized involution models of the projective reflection
groups $G(r,p,q,n)$. This family of groups parametrizes all quotients of the
complex reflection groups $G(r,p,n)$ by scalar subgroups. Our classification is
ultimately incomplete, but we provide several necessary and sufficient
conditions for generalized involution models to exist in various cases. In the
process we solve several intermediate problems concerning the structure of
projective reflection groups. We derive a simple criterion for determining
whether two groups $G(r,p,q,n)$ and $G(r,p',q',n)$ are isomorphic. We also
describe explicitly the form of all automorphisms of $G(r,p,q,n)$, outside a
finite list of exceptional cases. Building on prior work, this allows us to
prove that $G(r,p,1,n)$ has a generalized involution model if and only if
$G(r,p,1,n) \cong G(r,1,p,n)$. We also classify which groups $G(r,p,q,n)$ have
generalized involution models when $n=2$, or $q$ is odd, or $n$ is odd.
