We report a theoretical investigation of the electronic structure of the plutonium-based medium-high-${T}_{c}$ superconductor ${\mathrm{PuCoGa}}_{5}$, on the basis of ab initio local spin-density functional calculations. We furthermore report electronic structure calculations of the related actinide compounds ${\mathrm{PuRhGa}}_{5}$, ${\mathrm{PuIrGa}}_{5}$, ${\mathrm{UCoGa}}_{5}$, ${\mathrm{NpCoGa}}_{5}$, and ${\mathrm{AmCoGa}}_{5}$. ${\mathrm{PuRhGa}}_{5}$ is a superconductor as well, whereas the other materials do not become superconducting. The equilibrium lattice parameters within the tetragonal ${\mathrm{HoCoGa}}_{5}$ crystal structure are well reproduced for ${\mathrm{UCoGa}}_{5}$, ${\mathrm{NpCoGa}}_{5}$, as well as for the three isoelectronic $\mathrm{Pu}\text{\ensuremath{-}}115$ compounds when we assume delocalized $5f$ states. The possibility of a partial $5f$ localization occurring for the $\mathrm{Pu}\text{\ensuremath{-}}115$ compounds is discussed. The electronic structures of the three $\mathrm{Pu}\text{\ensuremath{-}}115$ compounds are computed to be rather similar: in each of the $\mathrm{Pu}\text{\ensuremath{-}}115$ materials the density of states at the Fermi energy is dominated by the $\mathrm{Pu}$ $5f$ contribution. Our total-energy calculations predict antiferromagnetic order to be favorable for all three $\mathrm{Pu}\text{\ensuremath{-}}115$ materials, which is, however, observed experimentally for ${\mathrm{PuIrGa}}_{5}$ only. Within the $\mathrm{Pu}\text{\ensuremath{-}}115$ series some small changes of the bands near the Fermi energy occur, which could be relevant for the superconductivity. A comparison of the ab initio calculated and the experimental properties clearly supports the picture of delocalized $5f$ electrons for ${\mathrm{UCoGa}}_{5}$. The neptunium-based 115 compound is predicted to order antiferromagnetically, which is supported by experiment. Also, the calculated magnetic moment $(0.85\phantom{\rule{0.3em}{0ex}}{\ensuremath{\mu}}_{B})$ compares well with the measured moment $(0.84\phantom{\rule{0.3em}{0ex}}{\ensuremath{\mu}}_{B})$. These findings advocate that the $\mathrm{Np}$ $5f$'s are still to some extent delocalized in ${\mathrm{NpCoGa}}_{5}$. In contrast, for the $\mathrm{Am}\text{\ensuremath{-}}115$ analog the $\mathrm{Am}$ $5f$ electrons can be expected to be localized. We furthermore discuss the theoretical Fermi surfaces and present calculated de Haas-van Alphen quantities for a comparison with future experiments. For ${\mathrm{UCoGa}}_{5}$ we obtain a semi-quantitative agreement with recently reported de Haas-van Alphen experiments. The possible origins of the superconductivity are discussed. Our investigation particularly reinforces the analogy to the heavy-fermion superconductors ${\mathrm{CeCoIn}}_{5}$ and ${\mathrm{CeIrIn}}_{5}$, however, with a stronger coupling strength due to a much stronger $5f$ hybridization.