Twisted Rabinowitz–Floer homology is the Morse–Bott homology in the sense of Floer associated with the twisted Rabinowitz action functional, generalising Rabinowitz–Floer homology introduced by Kai Cieliebak and Urs Frauenfelder in 2009. Using twisted Rabinowitz–Floer homology one can study periodic orbits in Hamiltonian systems admitting discrete symmetries. In this chapter we lay the foundations for constructing such a homology theory. A major obstacle to overcome is achieving compactness of the relevant moduli spaces, since one needs to bound Lagrange multipliers.