  We analyze the sensitivity of $\beta$-decay rates in 78 Ni and 100,132 Sn to
a correction term in Skyrme energy-density functionals (EDF) which modifies the
radial shape of the nucleon effective mass. This correction is added on top of
several Skyrme parametrizations which are selected from their effective mass
properties and predictions about the stability properties of 132 Sn. The impact
of the correction on high-energy collective modes is shown to be moderate. From
the comparison of the effects induced by the surface-peaked effective mass in
the three doubly magic nuclei, it is found that 132 Sn is largely impacted by
the correction, while 78 Ni and 100 Sn are only moderately affected. We
conclude that $\beta$-decay rates in these nuclei can be used as a test of
different parts of the nuclear EDF: 78 Ni and 100 Sn are mostly sensitive to
the particle-hole interaction through the B(GT) values, while 132 Sn is
sensitive to the radial shape of the effective mass. Possible improvements of
these different parts could therefore be better constrained in the future.
