I have encountered a situation that reminds me of holonomy/monodromy, but which takes place on a finite space. I am mostly looking for references, since this seems like a logical thing to investigate. I also have some other questions that might be answered here.
Let me set some notation. Let $G$ be a finite group, and $X$ a finite (principal) homogeneous space with respect to $G$. Let $S$ be a minimal generating set for $G$. Let $V$ be a (finite-dimensional) vector space, where in my case the underlying field is finite. $\textrm{Aut}(V)$ is the general linear group on $V$. Let $t$ be a map from $S$ to $\textrm{Aut}(V)$, such that the empty word maps to the identity element in $\textrm{Aut}(V)$, and $t(s^{-1}) = t(s)^{-1}$ for all $s\in S$.
Now, a path can be described by $w$ (a word in $S$), and a loop $l$ is then a word in $S$ that equals the identity element in $G$. With abuse of notation, set $t(w)=t(w_1w_2\ldots w_n) \equiv t(w_1)t(w_2)\ldots t(w_n)$ to be the ordered product of the images of the elements of the word $w$ under $t$.
The idea that $t(l)$ can be a non-identity element of $\textrm{Aut}(V)$ for a loop $l$ is how I interpret the notion of non-zero holonomy/monodromy. Two natural questions already arise.
First, is there a way to make sense out of the distinction between holonomy and monodromy? For the continuous case, (non-trivial) monodromy manifests itself from the fact that $t(l)$ is equal to the identity for homotopically trivial loops $l$ (and $t(l)$ not being the identity in general for homotopically non-trivial loops). Perhaps one could say that a loop is homotopically trivial iff it is equal to the identity using only the group axioms?
Second question, are there general statements one can make about the monodromy group that arises in this way? Any (mild) conditions on $G$, $t$ and $V$ would be fine with me.
One observation to make is that a Cayley graph of a finite group $G$ together with a representation $t: G\rightarrow \textrm{Aut}(V)$ gives a nice example of such a spaces. By moving around a loop in the graph, and picking up elements $t(g_i)$ for each generator /edge, we know that $t(l) $ needs to be the identity, leading to a 'flat' space.