In this paper we study the foundations of the algebraic treatment of classical and quantum field theories for Dirac fermions under external backgrounds following the initial contributions made by several collegues. The treatment is restricted to contractible spacetimes of globally hyperbolic nature in dimensions $d\ge 4$. In particular, we construct the classical M{\o}ller maps intertwining the configuration spaces of charged and uncharged fermions. In the last part, as a first step towards a quantization of the theory, we explore the combination of the classical M{\o}ller maps with Hadamard bidistributions and prove that they are involutive isomorphisms (algebraically and topologically) between suitable (formal) algebras of functionals (observables) over the configuration spaces of charged and uncharged Dirac fields.