Earth-like planets orbiting M dwarfs are prominent targets when searching for life outside the Solar System. We apply our Coupled Atmosphere Biogeochemical model to investigate the coupling between the biosphere, geosphere, and atmosphere in order to gain insight into the atmospheric evolution of Earth-like planets orbiting M dwarfs and to understand the processes affecting biosignatures and climate on such worlds. This is the first study applying an automated chemical pathway analysis quantifying the production and destruction pathways of molecular oxygen (O2) for an Earth-like planet with an Archean O2 concentration orbiting in the habitable zone of the M dwarf star AD Leonis, which we take as a type-case of an active M dwarf. The main production arises in the upper atmosphere from carbon dioxide photolysis followed by catalytic hydrogen oxide radical (HOx) reactions. The strongest destruction does not take place in the troposphere, as was the case in Gebauer et al. (2017) for an early Earth analog planet around the Sun, but instead in the middle atmosphere where water photolysis is the strongest. Results further suggest that these atmospheres are in absolute terms less destructive for O2 than for early Earth analog planets around the Sun despite higher concentrations of reduced gases such as molecular hydrogen, methane, and carbon monoxide. Hence smaller amounts of net primary productivity are required to oxygenate the atmosphere due to a change in the atmospheric oxidative capacity, driven by the input stellar spectrum resulting in shifts in the intrafamily HOx partitioning. Under the assumption that an atmosphere of an Earth-like planet survived and evolved during the early high-activity phase of an M dwarf to an Archean-type composition, a possible "Great Oxidation Event," analogous to that on Early Earth, would have occurred earlier in time after the atmospheric composition was reached, assuming the same atmospheric O2 sources and sinks as on early Earth. Key Words: Earth-like - Oxygen - M dwarf stars - Atmosphere - Biogeochemistry - Photochemistry - Biosignatures - Earth-like planets. Astrobiology 18, 856-872.