The Bauer–Furuta invariant, a stable homotopy theoretic refinement of the Seiberg–Witten invariant, is traditionally constructed using the so-called finite-dimensional approximations—an unnecessarily involved process requiring various auxiliary choices. In this talk, I will present a new framework based on the six-functor formalism for sheaves of spectra, which eliminates this reliance entirely. As such, this new approach ensures that the Bauer–Furuta invariant behaves naturally in families. I will also discuss its generalization to genuine equivariant sheaves on spaces with Lie group actions, where the equivariance of the Bauer–Furuta invariant arises naturally.