A fractional First Officer reported right engine overheating and cabin temperature malfunctions during climb; resulting in a diversion to a nearby airport.

Narrative

Climbing through approximately FL370 right engine Interstage Turbine Temperature (ITT) was increasing into yellow caution range. Captain reduced thrust lever setting on right side to lower ITT to a safe setting. I began to look in QRH for any guidance on the experienced engine issue; none was found and then Air Cycle Machine (ACM) overtemp and Emergency pressurization Crew Alert System (CAS) messages appeared. I began to run QRH for the ACM overtemp condition and waited for the CAS messages to go away while the plane was leveling off at FL410. Captain made a call to the company to discuss about best steps to take next; they decided on a diversion to ZZZ. Captain went to inform the passenger about the diversion and check on her as cabin temperatures were increasing from the emergency pressurization being active. I notified ATC of our intentions and followed their heading vectors and altitude changes. After the Captain returned I ran the emergency pressurization QRH as we descended towards ZZZ. The ACM overtemp condition did not clear and cabin temperatures continued to rise. Shortly after transferring to Approach Control they gave us a delay vector and then we informed that we needed no delays for landing. Approach Controller gave us priority handling to a visual approach for a landing on Runway XX. After touchdown I turned off the pressure source and bleeds in an attempt to reduce hot air inflow to the cabin. We rolled down the runway and took the taxiways into the ramp. After setting the parking brake I went to the rear to get passenger off the plane and cool down while waiting for the first responders to arrive.