Shape transitions of swirling flames are often observed in combustion chambers operated at steady flow injection conditions. These transitions may alter pollutant emissions, heat fluxes to the chamber walls or the system stability. In adiabatic combustors, transitions between V and M shapes of swirling flames are controlled by the dynamics of stretched flame elements propagating through the reactants diluted by the burnt products in the outer shear layer of the swirling combustible jet in contact with the hot outer recir- culation zone. The impact of H2 concentration in the fuel and of heat loss at the chamber walls modifying the temperature in the outer recirculation zone are analyzed in this study for lean premixed CH4/H2/air flames. Laser induced OH fluorescence in different planes and particle imaging velocimetry are used to determine the probability of V to M shape transitions when the hydrogen concentration or the temperature of the combustor walls are modified. The wall temperatures are determined by laser induced phosphores- cence and the temperature in the outer recirculation zone is inferred from thermocouple measurements. It is found that the probability of stabilizing a M flame increases with the H2 concentration in the combus- tible mixture.. For the same laminar burning velocity, the flame featuring a higher extinction limit to strain rate has a higher probability to take a M shape. When combustion is initiated with cold chamber walls, the V to M flame transition probability is reduced compared to a situation with hot walls. During thermal transient and at steady state, it is shown that the temperature of the combustion chamber walls is an important factor controlling these shape transitions due to its impact on the temperature of the burnt gases in the outer recirculation zone. This study confirms that the topology of swirling flames is highly sensitive to heat transfer to the combustion chamber walls.