The J/$\psi$ production is sensitive to the presence of the deconfined state of quarks and gluons, quark-gluon plasma (QGP), which is expected to form in ultrarelativistic nuclear collisions. Measurements of non-prompt J/$\psi$ originating from the weak decays of b-hadrons, can provide an estimate of beauty quark production in nuclear collisions. It is observed that the production of non-prompt J/$\psi$ is modified in Pb$-$Pb collisions in comparison to that in pp collisions at the same energy scaled by the number of binary collisions, as quantified by nuclear modification factors $(R_{\rm AA})$. It is related to hot medium effects taking place during the QGP phase. The measurement of nuclear modification factors in p$-$Pb collisions is used to assess the so-called Cold Nuclear Matter (CNM) effects which can further modify the production yields of non-prompt J/$\psi$ in collisions involving heavy-ions. The ALICE detector has excellent capabilities to measure J/$\psi$ in the $e^+e^-$ decay channel at midrapidity down to zero transverse momentum $(p_{\rm T})$ allowing the statistical separation of the non-prompt J/$\psi$ component for $p_{\rm T}$ down to $p_{\rm T}$ = 1 GeV/$c$.



In this contribution \textendash , ALICE midrapidity results on non-prompt J/$\psi$ production cross sections in pp collisions are presented and compared with the theoretical models. Moreover, $R_{\rm AA}$ of non-prompt J/$\psi$ at midrapidity as a function of $p_{\rm T}$ in p$-$Pb collisions at the center-of-mass energy per nucleon pair $\sqrt{s_{\rm NN}}$ = 5.02 TeV are presented and further compared with $R_{\rm AA}$ of non-prompt J/$\psi$ in Pb$-$Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV. Furthermore, results are compared with theoretical predictions.