ag.algebraic geometry - etale fundamental group and etale cohomology of curves

Given a curve $C$. Is there any relation between the etale fundamental group $pi_1(C)$ and the first etale cohomology of the constant sheaf , say $Z/nZ$, on $C$ ?

For example, if $C$ is a complex curve, then the singular cohomology $H^1(C,Z)$ is the dual of the topological fundamental group divided by the commutators ( which is the same as Hom$(pi_1(C),Z) )$.

So it seems that there should be some relation between Hom$(pi_1(C),Z/nZ)$ and $H^1(C,Z/nZ)$ in the etale case, but how?