Step 1: Understanding the Concept:
Heatstroke is a severe medical condition where the body's internal temperature rises above $104^{\circ}$F ($40^{\circ}$C).
Step 2: Detailed Explanation:
(A) Runny nose/coughing: Common for respiratory infections, not heatstroke.
(B) Rapid pulse and flushed skin: When the body overheats, the heart pumps faster to try to cool the body down, and blood vessels dilate, causing redness (flushing). This is a definitive sign of heatstroke.
(C) Cold/shivering: These are signs of hypothermia (cold exposure).
(D) Nausea/vomiting: These occur in heatstroke, but are very common in many other conditions (like food poisoning), making (B) a more specific physical sign.
Step 3: Final Answer:
Option (B) represents the physiological response to extreme heat.