  We discuss a $\sim 3\,\sigma$ signal (local) in the light Higgs-boson search
in the diphoton decay mode at $\sim 96$ GeV as reported by CMS, together with a
$\sim 2\,\sigma$ excess (local) in the $b \bar b$ final state at LEP in the
same mass range. We interpret this possible signal as a Higgs boson in the 2
Higgs Doublet Model type II with an additional Higgs singlet, which can be
either complex (2HDMS) or real (N2HDM). We find that the lightest CP-even Higgs
boson of the two models can equally yield a perfect fit to both excesses
simultaneously, while the second lightest state is in full agreement with the
Higgs-boson measurements at $125$ GeV, and the full Higgs-boson sector is in
agreement with all Higgs exclusion bounds from LEP, the Tevatron and the LHC as
well as other theoretical and experimental constraints. We derive bounds on the
2HDMS and N2HDM Higgs sectors from a fit to both excesses and describe how this
signal can be further analyzed at future $e^+e^-$ colliders, such as the ILC.
We analyze in detail the anticipated precision of the coupling measurements of
the $96$ GeV Higgs boson at the ILC. We find that these Higgs-boson
measurements at the LHC and the ILC cannot distinguish between the two
Higgs-sector realizations.
