PA-28-160 pilot reported engine sputtering while climbing to cruise altitude. The pilot returned to the departure airport and landed safely.

Narrative

Started preflight at about XA:50 am where I performed a walk around beginning with checking that both fuel tanks were topped off. Next I sumped the tanks and did not see any water so I put fuel back into tanks. Continuing the preflight; I performed a pre start on the engine and turned on electric fuel pump. The fuel system read 5 psi and I shut off fuel pump. Let engine warm up leaned out for taxi. Listened to ATIS set altimeter to 29.88 IN. Contacted Ground for taxi instruction and flight following with information was given frequency XXX.X and squawk XXXX. Went to run up area performed run up with check list perform mag drops no more than 20 rpms on left and right. Checked carb heat all other indications were in the green range. Contacted Tower was number two for take-off; told to taxi to XX left and call Tower. Enter Runway XX left on take-off roll the engine was running smoothly; all gauges were in the green range. Climbed out at 79 kts. over ZZZ1 and switched to ZZZ Approach requested 6500 ft. for cruise altitude. Continued flight cruise climb with engine full rich electric fuel pump on. Set compass bug onto 275 for direct to ZZZ2. At XB:20am 5000 ft. engine started sputtering applied carb heat with no change in sputtering [requested priority handling] gave fuel and souls on board diverted to ZZZ and landed long on XX left. was able to taxi to the parking and shut down.'