  We develop a fully self-consistent subtracted second random-phase
approximation for charge-exchange processes with Skyrme energy-density
functionals. As a first application, we study Gamow-Teller excitations in the
doubly-magic nucleus $^{48}$Ca, the lightest double-$\beta$ emitter that could
be used in an experiment, and in $^{78}$Ni, the single-beta-decay rate of which
is known. The amount of Gamow-Teller strength below 20 or 30 MeV is
considerably smaller than in other energy-density-functional calculations and
agrees better with experiment in $^{48}$Ca, as does the beta-decay rate in
$^{78}$Ni. These important results, obtained without \textit{ad hoc} quenching
factors, are due to the presence of two-particle -- two-hole configurations.
Their density progressively increases with excitation energy, leading to a long
high-energy tail in the spectrum, a fact that may have implications for the
computation of nuclear matrix elements for neutrinoless double-$\beta$ decay in
the same framework.
