  By using a non-self-consistent many-body $T$-matrix theory, we calculate the
finite-temperature Raman spectroscopy of a mobile impurity immersed in a Fermi
bath in three dimensions. The dependences of the Raman spectrum on the
transferred momentum, temperature, and impurity-bath interaction are discussed
in detail. We confirm that the peak in the Raman spectrum shows a weaker
dependence on the impurity concentration than that in the radio-frequency
spectroscopy, due to the nonzero transferred momentum, as anticipated. We
compare our theoretical prediction with the recent measurement by Gal Ness
\textsl{et al.} in Physical Review X \textbf{10}, 041019 (2020) without any
adjustable parameters. At weak coupling, we find a good quantitative agreement.
However, close to the Feshbach resonance the agreement becomes worse. At strong
coupling, we find that an unrealistic Fermi bath temperature might be needed,
in order to account for the experimental data.
