**Core Concept**
The cremasteric muscle is a part of the spermatic cord that plays a crucial role in regulating testicular temperature. It is innervated by a branch of the genitofemoral nerve, which is responsible for controlling the muscle's contraction and relaxation.

**Why the Correct Answer is Right**
The cremasteric muscle is supplied by the genital branch of the genitofemoral nerve. This branch arises from the genitofemoral nerve, which is a mixed nerve that originates from the lumbar plexus. The genital branch is responsible for innervating the cremaster muscle, as well as the cremasteric fascia and the dartos muscle. The contraction of the cremaster muscle is stimulated by the genital branch, causing the testes to elevate or descend in response to changes in ambient temperature.

**Why Each Wrong Option is Incorrect**
**Option B:** The femoral branch of the genitofemoral nerve primarily supplies the skin and fascia of the femoral region, and is not involved in the innervation of the cremaster muscle.
**Option C:** The lateral femoral nerve is a branch of the femoral nerve, which is responsible for innervating the muscles and skin of the anterior thigh. It is not involved in the innervation of the cremaster muscle.
**Option D:** The ilio-inguinal nerve is a branch of the lumbar plexus that primarily supplies the skin and muscles of the inguinal region, and is not involved in the innervation of the cremaster muscle.

**Clinical Pearl / High-Yield Fact**
The cremasteric reflex is an important clinical sign that involves the contraction of the cremaster muscle in response to stroking the skin of the upper inner thigh. This reflex is used to assess the integrity of the genitofemoral nerve and the cremaster muscle, and is often used in the diagnosis of testicular torsion or other conditions affecting the spermatic cord.

**✓ Correct Answer: A. Genital branch of genitofemoral nerve**