Large eddy simulation (LES) paradigms are employed to analyse the internal flow field of a lean premixed swirl-stabilized combustor with axial air injection at both non-reacting and reacting conditions, for a methane and a methane-hydrogen fuel mixture. The thickened flame combustion model (TFM) with detailed chemical kinetic mechanism is employed to simulate the flow. An adaptive mesh strategy is used to maximise the mesh resolution in the flame and boundary layer regions. The numerical results for the methane flame are firstly validated against experimental velocity measurements obtained via particle image velocimetry (PIV). Subsequently the LES is employed to simulate hydrogen-enriched methane flames by keeping the same output power in the combustor, in order to obtain insights on the flow behaviour when hydrogen is added, in terms of flame stability and emissions. A POD analysis reveals the presence of a precessing vortex core (PVC) in both reacting and non-reacting conditions, and how this PVC is affected by the reactants mixture is discussed in the paper. Moreover, the flame is observed to propagate upstream in the jet core despite the use of axial air injection, although flashback is not observed. In terms of emissions, significant reduction in CO and NO_x is observed when adding the hydrogen to the reactants mixture despite the higher flame speed, the reason for are discussed in the paper.