DAB TRACON Controller reported providing assistance to an aircraft after F11 TRACON refused to issue the aircraft an IFR clearance.

Narrative

Aircraft X contacted DAB South Radar sector looking for an IFR clearance. They were above a layer but wanted to have an IFR clearance and Air Traffic Services for safety. F11/Central FL TRACON had refused to give them an IFR clearance; and told them to contact FSS. The pilot was unable to; but was told to try DAB." I called the Operations Supervisor (OS)/Controller in Charge (CIC) position at F11 explaining what had happened and asked if they would be able to provide this aircraft with air traffic service. The OS/CIC said to switch the aircraft back; which we did. Shortly after the aircraft came back to DAB saying F11 would not help them. DAB Controller was able to issue the IFR clearance; and point the aircraft out to F11 still 20 miles in their airspace. Previous event reported involved F11 refusing service to a pair of aircraft unless 'they could use VHF frequency' (landing ZZZ). This is lazy at best - it's a severe safety issue that needs to be addressed swiftly with consequences."