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The constrained Hartree-Fock-Bogoliubov approximation, based on the recent parametrization D1M$^{*}$ of the Gogny energy density functional, is used to describe fission in 435 superheavy nuclei. The Gogny-D1M$^{*}$ parametrization is benchmarked against available experimental data on inner and second barrier heights, excitation energies of the fission isomers and half-lives in a selected set of Pu, Cm, Cf, Fm, No, Rf, Sg, Hs and Fl nuclei. Results are also compared with those obtained with the Gogny-D1M energy density functional. A detailed study of the minimal energy fission paths is carried out for isotopic chains with atomic numbers 100 $\le$ Z $\le$ 126 including very neutron-rich sectors up to around 4 MeV from the two-neutron driplines. Single-particle energies, ground state deformations, pairing correlations, two-nucleon separation energies and barrier heights are also discussed. In addition to fission paths, the constrained Hartree-Fock-Bogoliubov framework provides collective masses and zero-point quantum rotational and vibrational energies. Those quantities are building blocks within the Wentzel-Kramer-Brillouin formalism employed to evaluate the systematic of the spontaneous fission half-lives t$_\mathrm{SF}$. The competition between spontaneous fission and $α$-decay is studied, through the computation of the $α$-decay half-lives t$_\mathrmα$ using a parametrization of the Viola-Seaborg formula. From the comparison with the available experimental data and the results obtained with other theoretical approaches, it is concluded that D1M$^{*}$ represents a reasonable starting point to describe fission in heavy and superheavy nuclei.