  We discuss a simple renormalisable, gauge invariant model with a fermiophobic
$Z'$ boson: it has no couplings to the three Standard Model (SM) chiral
families, but does couple to a fourth vector-like (VL) family. The SM Higgs
couples to the fourth VL lepton, leading to an enhanced contribution to the
muon anomalous magnetic moment $\left(g-2\right)_{\mu}$. The latter
contribution requires a non-vanishing coupling of $Z'$ to right-handed muons,
which arises within this model due to mixing effects between the SM and VL
fermions, along with $Z'$ couplings to the second generation SM lepton doublet
and third generation SM quark doublet. This model can simultaneously account
for the measured $B$-decay ratios $R_{K^{(*)}}$ and $\left(g-2\right)_{\mu}$.
We identify the parameter space where this explanation is consistent with
existing experimental constraints coming from $B_{s}-\bar{B}_{s}$ mixing,
neutrino trident production and collider searches. We also check that the SM
Higgs coupling to the fourth VL lepton does not produce a dangerous
contribution to the Higgs diphoton decay.
