  We prove that a finite complex reflection group has a generalized involution
model, as defined by Bump and Ginzburg, if and only if each of its irreducible
factors is either $G(r,p,n)$ with $\gcd(p,n)=1$; $G(r,p,2)$ with $r/p$ odd; or
$G_{23}$, the Coxeter group of type $H_3$. We additionally provide explicit
formulas for all automorphisms of $G(r,p,n)$, and construct new Gelfand models
for the groups $G(r,p,n)$ with $\gcd(p,n)=1$.
