Zeroth cohomology of manifold with $\mathbb Z_2$ coefficients

Question

Let $M$ be a smooth manifold. Then the zeroth de Rham cohomology $$H_{\text{dR}}^0(M,\mathbb R)\cong \mathbb R^{\pi_0(M)}$$ counts the number of connected components of $M$. What does the 0-th cohomology $H^0(M,\mathbb Z_n)$ with coefficients in the cyclic group $\mathbb Z_n$ count?