  The Fermi constant ($G_F$) is extremely well measured through the muon
lifetime, defining one of the key fundamental parameters in the Standard Model
(SM). However, to search for physics beyond the SM (BSM), it is the precision
of the second-best independent determination of $G_F$ that defines the
sensitivity. The best alternative extractions of $G_F$ proceed via the global
electroweak (EW) fit or from superallowed $\beta$ decays in combination with
the Cabibbo angle measured in kaon, $\tau$, or $D$ decays. Both variants
display some tension with $G_F$ from muon decay, albeit in opposite directions,
reflecting the known tensions within the EW fit and hints for the apparent
violation of CKM unitarity, respectively. We investigate how BSM physics could
bring the three determinations of $G_F$ into agreement using SM effective field
theory and comment on future perspectives.
