  The study of the properties of the observed Higgs boson is one of the main
research activities in High Energy Physics. Although the couplings of the Higgs
to the weak gauge bosons and third generation quark and leptons have been
studied in detail, little is known about the Higgs couplings to first and
second generation fermions. In this article, we study the charm quark Higgs
coupling in the so-called $\kappa$ framework. We emphasize the existence of
specific correlations between the Higgs couplings that can render the measured
LHC Higgs production rates close to the SM values in the presence of large
deviations of the charm coupling from its SM value, $\kappa_c = 1$. Based on
this knowledge, we update the indirect bounds on $\kappa_c$ through a fit to
the precision Higgs measurements at the LHC. We also examine the limits on
$\kappa_c$ arising from the radiative decay $H \to J/\psi + \gamma$, the charm
quark-associated Higgs production, charm quark decays of the Higgs field and
charge asymmetry in $W^{\pm} + H$ production. Estimates for the future LHC
sensitivity on $\kappa_c$ at the high luminosity run are provided.
