I recommend saving the best for last, São Miguel, Azores.
Sintra is worth 2 days if you blitz through all there is to see in Rick’s guidebook, and you will be plenty tired. Near the train station is Dom Pipas. I have eaten there twice and would say it is one of the best restaurants I have been to in Portugal. Just Quinta da Regaleira is worth an entire morning or afternoon. To make life easier, do Sintra in the beginning or middle of your stay on the mainland, meaning Lisbon. I aways rent a car, but the guidebook and others on the forum will attest how easy it is to get to Sintra without a car. No car is necessary in Lisbon. Uber and taxis are plentiful and low cost.
Lisbon could keep you busy for months, and again, the guidebook offers plenty to do with 4 days. I used and thoroughly enjoyed Eco Tuk Tours in Lisbon. They can pick you up and drop you off at your hotel for a few extra euros. You will have a blast whether taking a company designed tour or make your own suggestion. It is a nice morning option to get your bearings. Lisbon offers almost limitless things to see and do.
So regarding São Miguel. A week is a wonderful time frame to truly explore this island paradise. I promise that your entire group will be amazed at what awaits you there.
Please make sure to rent, at a minimum, a 7 passenger van that will accommodate your group and probably most, if not all, your luggage. If you need to put all the possible extra luggage in a taxi, it is only about 10 euros to anywhere in Ponta Delgada, if that is where you are staying, and then you have the van for the week. When you fly out, again, if you need the taxi for the luggage, it can easily be arranged for the same price.
Try to do at least 1 hike on the island. There are dozens to choose from, depending on your comfort level. September should have spectacular weather for any beach time, but know that the water is not very warm. Locals will certainly be in the water. Your sightseeing days options are immense, and the nights will be enchanting, especially wandering around Ponta Delgada. I will be doing all of this in late July and early August!
At the top of this Portugal forum, the webmaster wrote a post regarding Azores advice. There is a link to the freemium section of my Azores guidebook. It has plenty of information for the Azores portion of your trip. I am certain that the freemium will give you lots of stress reducing ideas and planning tools, and it really is FREE :-).
I hope your group has a memorable trip. Send me a private message if there is anything I can help with in planning.
Carl