  This paper is a continuation of two previous papers in MSJ Memoirs, Vol.\,29
(Math. Soc. Japan, 2013) with the same title and numbered as I and II. Based on
the hereditary property given there, from mother groups $G(m,1,n)$, the
generalized symmetric groups, to child groups $G(m,p,n)$, the complex
reflection groups, we study in detail classification and construction of
irreducible projective representations (= spin representations) and their
characters of $G(m,1,n)$ for $n$ finite. Then, taking limits as $n$ tends to
infinity, we obtain spin characters of the inductive limit groups
$G(m,1,\infty)$. By the heredity studied further, this gives the main kernel of
the results for $G(m,p,\infty)$ with $p|m, p>1$.
