B767 First Officer reported a Cabin Pressure Warning on climb out as well as difficulties with using the Crew O2 Mask Microphones. The flight crew elected to perform an air turn back and precautionary landing.

Narrative

On the evening of DATE; I was the FO (First Officer) on a flight from ZZZ [to] ZZZ1. About 30 minutes after level off at FL380; the Cabin Altitude Warning lights; horn; and EICAS message were activated. Cabin altitude was around 11;000 [to] 12;000 ft. We accomplished the memory items and the QRH procedures and returned to ZZZ. One issue that I'd like to highlight; we were in Aircraft X; whose O2 system automatically triggers the microphone in the oxygen mask when you pull the mask out of its storage area. And the microphone in the mask stays active until you re-stow the mask. Re-stowing the mask at night was a little challenging; and we had to use the mask awhile for communication; even after we no longer needed it for oxygen. I plan to practice stowing the mask a little more in the future.After we landed in ZZZ; we wrote up the cabin pressure issue. MX (Maintenance) deferred the auto controller we initially used (Number 1) and we again departed. We paid careful attention to the cabin altitude during climb and cruise. At FL380; the pressure kept rising slowly and the differential was less than 8. At about 9;300 ft. cabin altitude; I had the Captain request FL340 (I was flying) to hopefully avoid another issue. After leveling at FL340; the Captain called [Dispatch] and MX to discuss the issue. We had plenty of fuel; so the lower altitude wouldn't be an issue. The cabin altitude responded much better at FL340; staying below 8;000 ft. As the flight progressed; the differential increased and the cabin altitude decreased to around 5;000 ft. Because the system didn't exceed any parameters like the previous flight; we didn't make any additional logbook entries. After the flight; the Captain called MX and briefed them fully on our observations. They understood; and let us know that the plane wasn't going back out any time soon.