Artificial intelligence (AI) models to automatically classify skin lesions from dermatology images have shown promising performance but also susceptibility to bias by skin tone. The most common way of representing skin tone information is the Fitzpatrick Skin Tone (FST) scale. The FST scale has been criticised for having greater granularity in its skin tone categories for lighter-skinned subjects. This paper conducts an investigation of the impact (on performance and bias) on AI classification models of granularity in the FST scale. By training multiple AI models to classify benign vs. malignant lesions using FST-specific data of differing granularity, we show that: (i) when training models using FST-specific data based on three groups (FST 1/2, 3/4 and 5/6), performance is generally better for models trained on FST-specific data compared to a general model trained on FST-balanced data; (ii) reducing the granularity of FST scale information (from 1/2 and 3/4 to 1/2/3/4) can have a detrimental effect on performance. Our results highlight the importance of the granularity of FST groups when training lesion classification models. Given the question marks over possible human biases in the choice of categories in the FST scale, this paper provides evidence for a move away from the FST scale in fair AI research and a transition to an alternative scale that better represents the diversity of human skin tones.