  Neutrino-nuclear responses in the form of squares of nuclear matrix
elements,NMEs, are crucial for studies of neutrino-induced processes in nuclei.
In this work we investigate magnetic hexadecapole (M4) NMEs in medium-heavy
nuclei. The experimentally derived NMEs, $M_{\rm EXP}$(M4), deduced from
observed M4 $\gamma $ transition half-lives are compared with the
single-quasiparticle (QP) NMEs, $M_{\rm QP}$(M4), and the microscopic
quasiparticle-phonon model (MQPM) NMEs $M_{\rm MQPM}$(M4). The experimentally
driven M4 NMEs are found to be reduced by a coefficient $k \approx 0.29 $ with
respect to $M_{\rm QP}$(M4) and by $k \approx 0.33$ with respect to $M_{\rm
MQPM}$(M4). The M4 NMEs are reduced a little by the quasiparticle-phonon
correlations of the MQPM wave functions but mainly by other nucleonic and
non-nucleonic correlations which are not explicitly included in the MQPM. The
found reduction rates are of the same order of magnitude as those for magnetic
quadrupole $\gamma $ transitions and Gamow-Teller (GT) and spin-dipole (SD)
$\beta $ transitions. The impact of the found reduction coefficients on the
magnitudes of the NMEs involved in astroneutrino interactions and neutrinoless
double beta decays are discussed.
