Continuous actions of topological groups on compact Hausdorff spaces X are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows one to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For Lyapunov stable actions the derived invariant mean $M\left({\varphi }_x\right)$ is continuous on X for any ϕ ∈ C(X), and the induced C*-valued...