**Core Concept**
The question requires the application of dermatological knowledge to diagnose a skin condition in an infant, considering the clinical presentation and family history. The key concept here is to differentiate between common skin conditions that can present with similar symptoms in infants, such as **scabies**, **infantile eczema**, **infantile seborrhoeic dermatitis**, and **impetigo contagiosa**.

**Why the Correct Answer is Right**
Scabies is a highly contagious skin infestation caused by the mite **Sarcoptes scabiei var. hominis**. The presence of multiple papules, exudative lesions, and vesicles, especially on the palms and soles, is characteristic of scabies in infants. The fact that the infant's mother had a history of itchy lesions further supports this diagnosis, as scabies is known to cause intense itching and can be spread through close contact. The distribution of lesions and the family history of itchy lesions are key factors that point towards scabies.

**Why Each Wrong Option is Incorrect**
**Option B:** Infantile eczema (atopic dermatitis) typically presents with dry, itchy skin and is often associated with other atopic conditions like asthma or allergies, but it doesn't usually involve the palms and soles in the early stages.
**Option C:** Infantile seborrhoeic dermatitis usually presents with a greasy, yellowish scaling on the scalp (cradle cap), face, and diaper area, which doesn't match the description given.
**Option D:** Impetigo contagiosa is a bacterial infection of the skin that presents with honey-colored crusts and is highly contagious, but it doesn't typically cause the widespread distribution of lesions or the specific involvement of palms and soles as described.

**Clinical Pearl / High-Yield Fact**
It's crucial to remember that scabies in infants can present differently than in adults, often with more widespread lesions and involvement of the palms and soles, and a family history of itchy lesions can be a significant clue. Early diagnosis and treatment are essential to prevent the spread of scabies.

**✓ Correct Answer: A. Scabies**