You can get fans with built-in humidistats or overrun timers or thermostats or wired to any sort of external device- but they're more expensive than basic fans, humidistats don't work that well in this country, timer is something else to go wrong & v annoying when you go to the loo at 3am and the fan whirrs on for another half an hour. If the tenant is paying leccie bill keep it simple- do check it is ducted as above and do clean it (and duct) regularly to combat fluff/gunge build-up
My humidistat fan in my upstairs bathroom is great. When I get up in the morning and run the bath it kicks in as required and stops when it should. From memory it was about £70. Admittedly, the humidistat added about £30 to the cost but, IMO, it was worth it.
At the time, I didn't want a fan that would kick in at 3AM when you turn the lights on to go for a pee and I was aware that I would not turn the lights on when using the bath or shower when the sun is up.
The fan in my downstairs bathroom is a larger capacity Icon with a motion sensor and humidistat. From memory, that was about £130. I think the PIR sensor sees dead people. It will start running at random times of the day.
That said, I did fit a humidistat fan for a customer that would kick in if it was raining outside. It was a PITA, the adjustment was wayyy to sensitive.