EMB-505 First Officer reported an Auto Pressurization Fail annunciation just after takeoff. Crew elected to return to departure airport for a normal landing.

Narrative

We departed ZZZ with 5 passengers en route to ZZZ1 in Aircraft X on Day 0. Upon take off (approximately 500' AGL); we received a Auto Pressurization Fail Crew Alert System (CAS) message. We continued our initial climb to 4;000' MSL as instructed by ATC. Upon reaching 4;000' we informed ATC that we needed to maintain 4;000' feet and troubleshoot our CAS message - they wanted us to continue our climb per the departure. After a little back and forth; they stated we could only remain at 4;000' if they [requested priority handling] for us. Being that we didn't want to risk further pressurization issues by going higher; we obliged and they provided vectors for us at 4;000'. We ran the QRH and agreed the flight could not be completed as planned. My PIC called the company via phone and discussed our options. Ultimately; the company recommended we return to ZZZ as they had another airplane ready for our passengers. After about 30 minutes of vectoring to make our maximum landing weight; we landed uneventfully at ZZZ and were able to manually equalize the pressurization upon landing.