Had a very pleasant stay in Lipari in one of Denise's rental flats.
Really appreciated the flat's terrace with a view of the sea.
The flat was very clean, looked recent, and Denise was very helpful (high reactivity to respond to my messages, collected me at the port and dropped me off for my departure).
The location is convenient: close (less than 5 min walk) to the Canneto beach (pebbles) and there is a regular minibus service to Lipari, with bus stops by the beach. I did not have a vehicle and it wasn't an inconvenience.
Please just note that the kitchen, while furnished with a stove and a fridge, doesn't have any of the basic "consumables" (salt, pepper, oil, washcloth,...) but there is a supermarket nearby where you can get everything you need.
During my stay there was an issue with the wifi.
The weather in September is still really warm (30°C+), the sea is also quite warm. Just beware of jellyfishes, but the water is so clear that you can see them (once you know you have to look!).
All in all, I very much enjoyed my stay there and would definitely recommend it!