I have an existing extractor fan fitted directly above the shower in the bathroom. 

It's an older house (1970s) and the fan appears to extract air directly into the roof cavity.

At first I thought this must be bad practice, but further research suggested this is common in houses with titled roofs as they are considered ventilated spaces.

My problem is that the ceiling around the fan is damp and damaged. I'm unsure if the damage to the ceiling is because of the damp air in the bathroom, damp air from venting into the roof cavity or a possible leak or bad seal on the fan itself.

The fan is currently working and appears to be doing its job. We don't have any damp or mould issues in the bathroom apart from the damp area directly around the vent itself. 

I'm not a DIYer and am happy to pay a tradesperson, but I don't know who to call.

An electrician because the fan is electrical? A builder or gibstopper to check/repair the damaged ceiling? ...or some other option.

Any advice or insights are appreciated.