The possibility of producing a measurable long-lived dark $Z$ boson, that is assumed to mix kinetically with the hypercharge gauge boson in Higgs decays and to be produced also in Higgs decays through Higgs-to-dark-Higgs mixing, at the Large Hadron Collider (LHC) is investigated. Displaced dimuons in the final state are considered where each of the $Z$ and the dark $Z$ bosons decays directly to a dimuon. The total cross sections for the decay modes of interest as well as the decay widths and decay lengths are calculated to next-to-leading order (NLO) by using Monte Carlo (MC) simulation in the framework of {\textsc{MadGraph5}}\_aMC@NLO and compared to the available analytical calculations to leading order (LO). The sensitivity of the LHC in Runs 2 and 3 to such searches is discussed.