Final answer:
The patient's symptoms suggest she has xeroderma pigmentosum, which is caused by a defective gene responsible for DNA excision repair, leading to a high risk of skin cancer.
Step-by-step explanation:
The defective gene in the patient described, who exhibits symptoms of red, scaling skin with minimal sun exposure and rapidly enlarging nevi, is responsible for DNA excision repair, specifically nucleotide excision repair (NER). This is indicative of the condition known as xeroderma pigmentosum, where patients are unable to repair damage caused by UV light, such as the formation of thymine dimers. These dimers significantly distort the DNA structure, leading to an increased risk of skin cancer.