The fundamental group and the first cohomology group

By a finite group cover of G, we will mean an M -definable connected group H with a surjective M -definable map H → G, whose kernel K is finite. The kernel K must lie within the center of H, because the action of H on K by conjugation yields a necessarily-trivial homomorphism from the connected group H to the finite group of automorphisms of K. If πi : Hi → G are finite group covers of G for i = 1, 2, a morphism of finite group covers from H1 to H2 will be an M -definable group homomorphism f : H1 → H2 making the obvious diagram commute. This makes finite group covers into a category.