  The complex representation rings of finite groups are the fundamental class
of fusion rings, categorified by the corresponding fusion categories of complex
representations. The category of $\mathbb{Z}_+$-modules of finite rank over
such a representation ring is also semisimple.
  In this paper, we classify the irreducible based modules of rank up to 5 over
the complex representation ring $r(S_4)$ of the symmetric group $S_4$. Totally
16 inequivalent irreducible based modules are obtained. Based on such a
classification result, we further discuss the categorification of based modules
over $r(S_4)$ by module categories over the complex representation category
${\rm Rep}(S_4)$ of $S_4$ arisen from projective representations of certain
subgroups of $S_4$.
