Phosphoglucose isomerase is the enzyme responsible for converting glucose-6-phosphate to fructose-6-phosphate in glycolysis. Hexokinase is involved in the phosphorylation of glucose, aldolase catalyzes the cleavage of fructose-1,6-bisphosphate, and phosphofructokinase-1 is involved in another key step in glycolysis, not this conversion.