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Find two hard to reverse functions f and h such that f ∘ h ∘ g = h (f and h injective, no constraints on g)

I am looking from 3 functions $f,g,h$ from $\mathbb N \to \mathbb N$ (they can be bijections, they need to be injective at least), such that:

$$f \circ h \circ g = h $$

and $f$ is hard to reverse programmatically, and $h$ is hard to reverse programmatically.

Right now, I am using AES SP 801-108 for $h$ but any function hard to reverse will do. $g$ and $h$ must be public (no secret key).