The evolution of Population II Red Giants has been studied by adopting Reimers' formalism for the efficiency of mass loss. Evolutionary computations for models of 0.8M0, Y = 0.23 and Z = 0.0002 are presented for a wide range of values of the parameter TI governing the mass loss rate. We find that for values of TI greater than about 0.70 the decrease in the envelope mass forces the star to leave the RGB before undergoing the He flash. However the core evolution is not affected by the surface conditions until long after the RG phase, and models with TI less than 0.80 eventually succeed in igniting the helium on the blue side of the H-R diagram. The variation of sud a scenario with metallicity is also discussed. Finally, we analyze the production and cooling of He white dwarfs produced by the mass loss mechanism. Theoretical constraints concerning the possible occurrence of He dwarfs are discussed, showing that even a 10% of RG stragglers can give contribution to the white dwarfs' cooling sequence observed in clusters.