Double detonations in sub-Chandrasekhar mass carbon-oxygen white dwarfs with helium shell are a potential explosion mechanism for a Type Ia supernova. It comprises a shell detonation and subsequent core detonation. The focus of our study is on the effect of the progenitor metallicity on the nucleosynthetic yields. For this, we compute and analyse a set of eleven different models with varying core and shell masses at four different metallicities each. This results in a total of 44 models at metallicities between 0.01Z_{sun}_ and 3Z_{sun}_. Our models show a strong impact of the metallicity in the high density regime. The presence of ^22^$Ne causes a neutron-excess which shifts the production from ^56^Ni to stable isotopes such as ^54^Fe and ^58^Ni in the {alpha}-rich freeze-out regime. The isotopes of the metallicity implementation further serve as seed nuclei for additional reactions in the shell detonation. Most significantly, the production of ^55^Mn increases with metallicity confirming the results of previous work. A comparison of elemental ratios relative to iron shows a relatively good match to solar values for some models. Super-solar values are reached for Mn at 3Z_{sun}_ and solar values in some models at Z_{sun}_. This indicates that the required contribution of Type Ia supernovae originating from Chandrasekhar mass WDs can be lower than estimated in previous work to reach solar values of [Mn/Fe] at [Fe/H]=0. Our galactic chemical evolution models suggest that Type Ia supernovae from sub-Chandrasekhar mass white dwarfs, along with core-collapse supernovae, could account for more than 80% of the solar Mn abundance. Using metallicity-dependent Type Ia supernova yields helps to reproduce the upward trend of [Mn/Fe] as a function of metallicity for the solar neighborhood. These chemical evolution predictions, however, depend on the massive star yields adopted in the calculations.