Functional equations on finite groups of substitutions

Motivated by some investigations of Babbage, we study a class of single variable functional equations. These are functional equations involving one unknown function and a finite set of known functions that form a group under the operation of composition. It turns out that the algebraic structure of a stabilizer determines the number of initial value conditions for the functional equation. In the proof of the main result, the Implicit Function Theorem and, when the stabilizer is nontrivial, the Global Existence and Uniqueness Theorem play a key role.