Let $X$ be a noetherian scheme over a field, or any other topological space such that the following makes sense. Let $\mathcal F$ be a locally constant sheaf on $X$ (for the Zariski topology, or on the étale site...).
Is it possible to recover the set $\pi_0(X)$ of connected components of $X$ from the global sections $\mathrm{H}^0(X,\mathcal F)$ of $\mathcal F$ ?
Of course it is true when $\mathcal F$ is a constant sheaf, say $\mathcal F = \underline{A}$ where $A$ is a given set (group, module, ...). In this case, we have $\mathrm H^0(X,\underline{A}) = A^{\pi_0(X)}$.
I expect the answer to be negative in general, but I may lack some examples and in-depth understanding to get a feeling why.