  While the Hoyle state (the isoscalar $0^+_2$ excitation at 7.65 MeV in
$^{12}$C) has been observed in almost all the electron and $\alpha$ inelastic
scattering experiments, the second $2^+$ excited state of $^{12}$C at $E_{\rm
x}\approx 10$ MeV, believed to be an excitation of the Hoyle state, has not
been clearly observed in these measurements excepting the high-precision \aap
experiments at $E_\alpha=240$ and 386 MeV. Given the (spin and isospin zero)
$\alpha$-particle as a good probe for the nuclear isoscalar excitations, it
remains a puzzle why the peak of the $2^+_2$ state could not be clearly
identified in the measured \aap spectra. To investigate this effect, we have
performed a microscopic folding model analysis of the \ac scattering data at
240 and 386 MeV in both the Distorted Wave Born Approximation (DWBA) and
coupled-channel (CC) formalism, using the nuclear transition densities given by
the antisymmetrized molecular dynamics (AMD) approach and a complex CDM3Y6
density dependent interaction. Although AMD predicts a very weak transition
strength for the direct $(0^+_1\to 2^+_2)$ excitation, our detailed analysis
has shown evidence that a weak \emph{ghost} of the $2^+_2$ state could be
identified in the 240 MeV \aap data for the $0^+_3$ state at 10.3 MeV, when the
CC effects by the indirect excitation of the $2^+_2$ state are taken into
account. Based on the same AMD structure input and preliminary \aap data at 386
MeV, we have estimated relative contributions from the $2^+_2$ and $0^+_3$
states to the excitation of $^{12}$C at $E_{\rm x}\approx 10$ MeV as well as
possible contamination by $3^-_1$ state.
