Repair crumbling edges of deliberate (wanted) hole in plaster ceiling
Extractor fan types can vary widely. The type that you describe sounds like a terrible design. My suggestion is that you replace the unit with one of the type fans that consist of an in-ceiling rectangular enclosure that mounts to the ceiling framing material.
If properly installed the edges of the enclosure should fit down into the hole in the ceiling plaster board and come down just even with the ceiling surface. These units come with a baffle piece that attaches to the enclosure structure to hold it in place. Retainers for the baffle may screw on, clip in or use spring clips. A very common type uses spring clips that allow the baffle to be clipped into slots in the enclosure and as the baffle is pushed up into place the spring spreads some to hold the baffle firmly to the ceiling.
For your situation you should look for a unit that has a cutout hole size that is bigger than the current hole plus the damaged area around it. Depending upon the location of the framing members in the ceiling it may be needed to add an additional piece of blocking between two existing ceiling joists to provide a place to secure the fan enclosure. Actually the best situation to provide the most secure mount for the enclosure is to have there be framing material along two sides of the enclosure.
Using this technique will eliminate any repeated fraying of the plaster board edge. In addition the fact that the fan enclosure extends to the ceiling surface it helps to isolate the edges of the plaster board from the damp air that the fan draws up and out the vent pipe.