There is a question I would like to ask at MO, but it seems somewhat unorthodox compared most others, so I want to get some support (or discouragement) here first. What I want to do is to give a new definition, justify why it is a natural extension of a well known concept, and ask if someone has seen it or has something to say about it. I am trying to extend a theory that I understand to a setting outside my main expertise, and I therefore risk making a trivial question. I have put effort in trying to find research or notes about this new object but to no avail. I do have genuine research interest in this thing, so it is not only a matter of curiosity.

Would something like this make a good question? (I don't see why not, but I have not encountered such questions here yet, so I'm slightly worried.) Is there something that I should be particularly careful about when formulating a question of this kind? Are there examples of good and bad questions like this to learn from?

I chose to make this meta question quite general in order to focus discussion on the idea of definitions as questions rather than the particulars of the question I had in mind. But if you want a short description of my question, I can give one. The full question is lengthy, so I will not produce it here.

A short version of the question I had in mind:

Periodic geodesics on compact Lie groups can be described algebraically, without any reference to minimizing arc length: a periodic geodesic is a mapping $S^1\ni t\mapsto x\phi(t)\in G$ where $\phi:S^1\to G$ is a nontrivial homomorphism and $x\in G$. By analogue, we can define a geodesic on a finite group by replacing $S^1$ with a finite cyclic group. The geodesic flow on finite groups can be seen as a discrete time dynamical system.

Have geodesics on finite groups been studied before, perhaps under another name? Does this structure look familiar to anyone?A problem in the field of inverse problems asks whether a function on a closed manifold is uniquely determined by its integrals over all closed geodesics. This problem has been studied quite a lot, also on Lie groups. A natural generalization of the question now asks whether a function on a finite group is uniquely determined by its sums over all geodesics.Has this problem been studied before? Is it known to have applications, abstract or concrete?I have obtained some results on this finite generalization. I am fairly confident that this problem is new to the inverse problems community, but I am not sure if it is well known in another field. It would be great if I could motivate the question or relate it to existing literature. Any ideas, references or analogues could be helpful in understanding and solving the problem.