  In this paper we simultaneously explain the excesses of the 750 GeV diphoton,
muon g-2 and $h\to \mu\tau$ in an extension of the two-Higgs-doublet model
(2HDM) with additional vector-like fermions and a CP-odd scalar singlet ($P$)
which is identified as the 750 GeV resonance. This 750 GeV resonance has a
mixing with the CP-odd scalar ($A$) in 2HDM, which leads to a coupling between
$P$ and the SM particles as well as a coupling between $A$ and the vector-like
fermions. Such a mixing and couplings are strongly constrained by
$\tau\to\mu\gamma$, muon g-2 and the 750 GeV diphoton data. We scan over the
parameter space and find that such an extension can simultaneously account for
the observed excesses of 750 GeV diphoton, muon g-2 and $h\to \mu\tau$. The 750
GeV resonance decays in exotic modes, such as $P\to hA$, $P\to HZ$, $P\to HA$
and $P\to W^\pm H^\mp$, and its width can be dozens of GeV and is sensitive to
the mixing angle.
