We consider photoemission from the $4d$ shell of the free Xe and encapsulated Xe@${\mathrm{C}}_{60}$ atoms by ionizing XUV and probing IR fields typical for a RABBITT (reconstruction of attosecond beating by interference of two-photon transitions) measurement. Our theoretical model is based on the numerical solution of the time-dependent Schr\odinger equation in the single-active-electron approximation. The fullerene ${\mathrm{C}}_{60}$ cage is represented by a finite-width well potential. We test our model against an analogous set of nonrelativistic [Phys. Rev. A 89, 053424 (2014)] and relativistic [Phys. Rev. A 96, 053407 (2017)] calculations for the $4d$ shell of Xe and Xe@${\mathrm{C}}_{60}$ driven by a continuous XUV field. Based on this verification, we make predictions for the total ionization probability, angular anisotropy $\ensuremath{\beta}$ parameter, and the angular-dependent atomic time delay ${\ensuremath{\tau}}_{a}$ from the threshold to several hundred eV of excess energy.
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