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We investigate the automorphism group of finite Gödel algebras, the algebraic counterpart of Godel infinite-valued propositional logic with a finite number of variables. In logical terms, we look at the structure of substitution of terms that preserve logical equivalence in this logic. We obtain a characterisation of the arising automorphism groups in terms of semidirect and direct products of symmetric groups. Building on this, we establish an explicit closed formula for the cardinality of the automorphism group of the Lindenbaum algebra of Gödel logic over n propositional variables, for any integer n >= 1.