  The elastic scattering cross section measured at energies $E\lesssim 10$
MeV/nucleon for some light heavy-ion systems having two identical cores like
$^{16}$O+$^{12}$C exhibits an enhanced oscillatory pattern at the backward
angles. Such a pattern is known to be due to the transfer of the valence
nucleon or cluster between the two identical cores. In particular, the elastic
$\alpha$ transfer has been shown to originate directly from the core-exchange
symmetry in the elastic $^{16}$O+$^{12}$C scattering. Given the strong
transition strength of the $2^+_1$ state of $^{12}$C and its large overlap with
the $^{16}$O ground state, it is natural to expect a similar $\alpha$ transfer
process (or inelastic $\alpha$ transfer) to take place in the inelastic
$^{16}$O+$^{12}$C scattering. The present work provides a realistic coupled
channel description of the $\alpha$ transfer in the inelastic $^{16}$O+$^{12}$C
scattering at low energies. Based on the results of the 4 coupled
reaction-channels calculation, we show a significant contribution of the
$\alpha$ transfer to the inelastic $^{16}$O+$^{12}$C scattering cross section
at the backward angles. These results suggest that the explicit coupling to the
$\alpha$ transfer channels is crucial in the studies of the elastic and
inelastic scattering of a nucleus-nucleus system with the core-exchange
symmetry.
