  In the Ge-Sr mass region, isotopes with neutron number $N \leq 40$ are known
to feature rapid shape changes with both nucleon number and angular momentum.
To gain new insights into their structure, inelastic proton scattering
experiments in inverse kinematics were performed on the rare isotopes
$^{74,76}$Kr. This work focuses on observables related to the $J^{\pi} = 4^+_1$
states of the Kr isotopes and, in particular, on the hexadecapole degree of
freedom. By performing coupled-channels calculations, hexadecapole deformation
parameters $\beta_4$ were determined for the $J^{\pi} = 4^+_1$ states of
$^{74,76}$Kr from inelastic proton scattering cross sections. Two possible
coupled-channels solutions were found. A comparison to predictions from nuclear
energy density functional theory, employing both non-relativistic and
relativistic functionals, clearly favors the large, positive $\beta_4$
solutions. These $\beta_4$ values are unambiguously linked to the well deformed
prolate configuration. Given the $\beta_2 - \beta_4$ trend, established in this
work, it appears that $\beta_4$ values could provide a sensitive measure of the
nuclear shell structure.
