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Explicit Brauer Induction formulae with certain natural behaviour have been developed for complex representations, for example by work of Boltje, Snaith and Symonds. In this paper we present induction formulae for symplectic and orthogonal representations of finite groups. The problems are motivated by number theoretical and topological questions. We will prove naturality with respect to restriction and inflation. Also we investigate complexification maps and use them to compare the orthogonal and symplectic induction formulae with Boltje's complex induction formula.