  The present paper explores possible features of electron elastic scattering
off endohedral fullerenes $A$@C$_{60}$. It focuses on how dynamical
polarization of the encapsulated atom $A$ by an incident electron might alter
scattering off $A$@C$_{60}$ compared to the static-atom-$A$ case, as well as
how the C$_{60}$ confinement modifies the impact of atomic polarization on
electron scattering compared to the free-atom case. The aim is to provide
researchers with a "relative frame of reference" for understanding which part
of the scattering processes could be due to electron scattering off the
encapsulated atom and which due to scattering off the C$_{60}$ cage. To meet
the goal, the C$_{60}$ cage is modeled by an attractive spherical potential of
a certain inner radius, thickness, and depth which is a model used frequently
in a great variety of fullerene studies to date. Then, the Dyson equation for
the self-energy part of the Green's function of an incident electron moving in
the combined field of an encapsulated atom $A$ and C$_{60}$ is solved in order
to account for the impact of dynamical polarization of the encaged atom upon $e
+ A@{\rm C_{60}}$ scattering. The Ba@C$_{60}$ endohedral is chosen as the case
study. The impact is found to be significant, and its utterly different role
compared to that in $e + Ba$ scattering is unraveled.
