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Charmonium production has been widely recognized as an excellent probe for investigating the properties of hot and dense nuclear matter formed in ultrarelativistic nuclear collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) within the framework of Quantum Chromodynamics (QCD). The measurements of directly produced J/$\psi$ and those originated from feed-down of heavy charmonium states, prompt J/$\psi$, provide a direct comparison with models that include J/$\psi$ production through regeneration, the dominant production mechanism at low transverse momentum ($p_{\rm T}$) and in central collisions at the LHC. This is crucial to understand the J/$\psi$ production via regeneration mechanism. The measurements of J/$\psi$ originating from the weak decay of b-hadrons, non-prompt J/$\psi$, provide an estimate of beauty quark production in nuclear collisions. Such measurements are important to investigate the beauty quark energy loss in the nuclear medium. The ALICE experiment has unique tracking and particle identification capabilities down to very low momentum at midrapidity ($|y| <$ 0.9), enabling the J/$\psi$ reconstruction down to $p_{\rm T}$ $\sim$ 0 and separation of prompt and non-prompt J/$\psi$ down to $p_{\rm T}$ $\sim$ 1.5 GeV/$c$ in Pb-Pb collisions. This study utilizes the complete datasets from Pb-Pb collisions at center of mass energy, $\sqrt{s_{\rm NN}}$ = 5.02 TeV, collected by the ALICE experiment during the LHC Run 2 program. The analysis is performed by reconstructing J/$\psi$ meson at midrapidity in the dielectron decay channel. To separate the prompt and non-prompt J/$\psi$ contributions, non-prompt J/$\psi$ fraction ($f_{\rm B}$) is extracted by simultaneous unbinned fits on invariant mass and pseudoproper decay length of dielectron candidates in different $p_{\rm T}$ intervals in the range 1.5 $<$ $p_{\rm T}$ $<$ 10 GeV/$c$. Additionally, the analysis is performed in different centrality intervals, namely, 0-10%, $\text{10-30%}$, and 30-50%. The fits rely on the templates obtained from Monte-Carlo simulation and data collected in the experiment. The fractions obtained from the fits are corrected for the detector's acceptance and reconstruction efficiency. Similarly, non-prompt J/$\psi$ fractions in pp collisions are obtained by interpolating available non-prompt J/$\psi$ measurements at midrapidity in the same $p_{\rm T}$ intervals as in Pb-Pb collisions. The production yields of prompt and non-prompt J/$\psi$ are determined by scaling inclusive J/$\psi$ $R_{\rm AA}$ by the ratio of $f_{\rm B}$ in Pb-Pb to $f_{\rm B}$ in pp collisions in the same $p_{\rm T}$ and centrality intervals. Systematic uncertainties in $f_{\rm B}$ measurements in Pb-Pb collisions due to possible sources are estimated and propagated to the measurements of yields of prompt and non-prompt J/$\psi$. The prompt and non-prompt J/$\psi$ production is modified in nuclear collisions in comparison to binary nucleon-nucleon collisions at the same collision energy due to various nuclear effects. Such modifications can be quantified by measurements of nuclear modification factors ($R_{\rm AA}$). The nuclear modification factors of prompt and non-prompt J/$\psi$ are measured as a function of $p_{\rm T}$ in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV across aforementioned centrality intervals. The systematic uncertainties obtained in the $f_{\rm B}$ measurements are propagated to the measurements of the $R_{\rm AA}$. The final results are compared with the previous measurements performed by the ALICE experiment at lower collision energy $\sqrt{s_{\rm NN}}$ = 2.76 TeV. Notably, the presented new results are more precise in terms of $p_{\rm T}$ and centrality intervals compared to results at lower collision energy. The results are further compared with similar measurements at high $p_{\rm T}$ by the CMS and ATLAS experiments. The measurements presented in this study complement the existing results obtained at high $p_{\rm T}$ by extending the analysis to lower $p_{\rm T}$ values. The state-of-the-art theoretical model calculations that include several medium effects for charm and beauty quarks depending on $p_{\rm T}$ are adopted to compare with the production yields and $R_{\rm AA}$ of both prompt and non-prompt J/$\psi$. An extension to these studies is also presented, where the future plans of ALICE experiment for such measurements are briefly discussed.