Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $X_0$ with value $\Lambda$, an abelian group.
Clearly $\Lambda_n$ is the constant sheaf on $X_n$ with value $\Lambda$ too.
There are, then, transition maps $f_{n+1,n}^{-1}\Lambda_n\to\Lambda_{n+1}$.
A base for the topology on $X := \varprojlim_{n\ge 0}X_n$ is given by opens of the form $U:= \varprojlim_{m\in I}U_m$, $I\subset\mathbf{N}$, $U_m\subset f^{-1}_{m,k}(U_k)$ open, $m,k\in I$, and the assignment:
$$U\mapsto \varinjlim_{m\in I}\Lambda_m(U_m)$$
uniquely determines a sheaf $\mathcal{F}_{\Lambda}$ on $X$.
One can give $X$ the structure of a solenoid in the sense of Smale (see here and the book by Moore and Sochet).
Say $\Lambda = \mathbf{C}$ the sheaf of locally constant smooth functions with value in the complex numbers. On $X$ one can talk about the sheaf $\mathcal{F}’$ of continuous complex valued functions that are smooth along the “leaves” and continuous along the transversals (these are copies of the Cantor set).
Are $\mathcal{F}_{\Lambda}$ and $\mathcal{F}’$ the same sheaf on $X$? Do they at least have the same cohomology?
In case this question has a negative answer, then:
Is $\mathcal{F}_{\Lambda}$ the constant abelian sheaf on $X$ with value $\Lambda$?
I expect the answer to this last question is “no”.
Related: this seems a related question, esp the answer by Ian Agol.