Air carrier pilot reported a pressurization failure during initial climb and a diversion to a nearby airport.

Narrative

Assigned ZZZ1 - ZZZ2 for aircraft that was coming out of Maintenance. During takeoff we could feel the aircraft overpressurizing more intensely than normal. We continued a normal climb up to 3000 feet and completed the after takeoff checklist. ATC gave us a climb to 16;000 and we told them we needed to stay at 3000 to work out a mechanical issue. I transferred control of the aircraft and saw the aircraft cabin rate would climb to about 4000 ft per min (the maximum it can display) and above 3.0 differential pressure until there was a loud bang or pop coming from the rear of the aircraft; to where the cabin rate would immediately reverse to 4000 fpm down until it depressurized to about 0.5-1.0 differentual pressure; to where the cycle would start all over every 3-5 seconds approximately. There is no checklist for an overpressurization fault; and we had not reached the maximum differential pressure for the overpressure relief valve; so we opened the outflow valve per the 'cabin altitude uncontrollable' portion of the unscheduled pressurization change QRH to relieve physiological pressure and prevent a possible overpressurization should the valve not open on the ground. Being at 3000 ft; we determined it was not necessary or unsafe to be unpressurized. I continued with the with the non routine landing considerations QRH and let Dispatch know we would need to return; to which they determined ZZZ would be a more suitable diversion field. Landed without event and made appropriate logbook entry.