The correct answer is E. Cryoglobulins.
Pernio is characterized by a single or multiple erythematous, violaceous, and edematous papules, nodules, or plaques that presents 12-24 hours after exposure to non-freezing cold or wet environment. They can be associated with burning, pruritus, or pain and tenderness and can last 1-2 weeks. In some instances, they can form vesicles or ulcerate.In kids, pernio can be associated with cryoglobulins and cold agglutinins.
Lupus Osteoma cutis, diabetes, calcinosis cutis are not associated with pernio. Although, pernio may be secondary to connective tissue disease, such as lupus, in adult. It is not a common presentation in children.
References: Weston WL, Morelli JG. Childhood pernio and cryoproteins. Pediatr Dermatol. 2000;17(2):97-99. doi:10.1046/j.1525-1470.2000.01722.x