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 $f_{n+1,n}^{-1}\Lambda_n\to\Lambda_{n+1}$, and one can take the direct limit $\mathcal{F} := \varinjlim_{n\ge 0}\Lambda_n$, a sheaf on the inverse limit topological space $X := \varprojlim_{n\ge 0}X_n$.

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 $\mathcal{F}$ 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 $\mathcal{F}$ the constant abelian sheaf on $X$ with value $\Lambda$?

I expect the answer to be no to this one, though.

Related: this seems a related question, esp the answer by Ian Agol.