Insulating FeGa$_3$ poses peculiar puzzles beyond the occurrence of an electronic gap in an intermetallic compound. This Fe-based material has a very distinctive structural characteristic with the Fe atoms occurring in dimers. The insulating gap can be described comparably well in either the weakly correlated limit or the strongly correlated limit within density functional theory viewpoints, where the latter corresponds to singlet formation on the Fe$_2$ dimers. Though most of the calculated occupied Wannier functions are an admixture of Fe $3d$ and Ga $4s$ or $4p$ states, there is a single bonding-type Wannier function per spin centered on each Fe$_2$ dimer. Density functional theory methods have been applied to follow the evolution of the magnetic properties and electronic spectrum with doping, where unusual behavior is observed experimentally. Both electron and hole doping are considered, by Ge and Zn on the Ga site, and by Co and Mn on the Fe site, the latter introducing direct disturbance of the Fe$_2$ dimer. Results from weakly and strongly correlated pictures are compared. Regardless of the method, magnetism including itinerant phases appears readily with doping. The correlated picture suggests that in the low doping limit Mn (for Fe) produces an in-gap hole state, while Co (for Fe) introduces a localized electronic gap state.
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