  We study the $S3$ symmetric extension of the Standard Model in which all the
irreducible representations of the permutation group are occupied by $SU(2)$
scalar doublets, one of which is taken as inert and can lead to dark matter
candidates. We perform a scan over parameter space probing points against
physical constraints ranging from unitarity tests to experimental Higgs
searches limits. We find that the latter constraints severely restrict the
parameter space of the model. For acceptable points we compute the value of the
relic density of the dark scalar candidates and find that it has a region for
low dark matter masses which complies with the Higgs searches bounds and lies
within the experimental Planck limit. For masses $\gtrsim 80$ GeV the value of
the relic density is below the Planck bound, and it reaches values close to it
for very heavy masses $\sim 5$ TeV. In this heavy mass region, this opens the
interesting possibility of extending the dark sector of the model with
additional particles.
