What heat-related illness is characterized by a lack of increase in rectal temperature?

The condition characterized by a lack of increase in rectal temperature is indeed heat syncope. This illness typically results from a sudden drop in blood pressure and insufficient blood flow to the brain, often following prolonged standing or sudden changes in position. As a result, individuals may experience dizziness or fainting, but their body temperature remains normal because the body's heat regulation processes are still functioning effectively.

In contrast, heat exhaustion involves elevated body temperature along with symptoms like weakness, sweating, and dizziness, while heat stroke is a severe condition where the body temperature rises significantly, posing serious health risks. Heat cramps are localized muscle spasms that can occur due to electrolyte loss and dehydration during heat exposure, but they also don't cause a lack of temperature increase. Hence, heat syncope is distinct in that it does not result in an increase in rectal temperature, making it the correct answer to the question.