$\boldsymbol{(g-2)_{e,\mu}}$ in an extended inverse type-III seesaw.

There has been a long-standing discrepancy between the experimental measurements of the electron and muon anomalous magnetic moments and their predicted values in the Standard Model. This is particularly relevant in the case of the muon $g-2$, which has attracted a remarkable interest in the community after the long-awaited announcement of the first results by the Muon $g-2$ collaboration at Fermilab, which confirms a previous measurement by the E821 experiment at Brookhaven and enlarges the statistical significance of the discrepancy, now at $4.2 \sigma$. In this paper we consider an extension of the inverse type-III seesaw with a pair of vector-like leptons that induces masses for neutrinos at the electroweak scale and show that one can accommodate the electron and muon anomalous magnetic moments, while being compatible with all relevant experimental constraints.