So many things to consider: What time of year are you going? I will assume summer, so recommend starting in the south and working your way north due to weather (I hate the heat).
How do you plan to get from one place to another? With a group that size, I will assume it’s by train. Renting a vehicle large enough for 5 plus luggage could be a challenge. You don’t want to pick up ca car in one country and drop off in another- the fees can be up to $1000 to go so.
What do you want to see? Big cities? Small villages? Mixture of both?
We did a 3 week road trip with our two teens in 2018. Started in Frankfurt, then on to Boppard on the Rhine, then Rothenburg, then Salzburg, Ljubljana, Venice, ended in Munich. Our focus was smaller towns, beautiful scenery, castles, WWII history, fried food and ice cream.
Apartments will be the way to go. Finding actual beds for 5 people may be a challenge. My kids don’t do sofa beds well. Many apartments that are “2 bedroom” consider a sofa bed a “bed room”.
Laundry will be something to consider. Another reason we like apartments. Just know that many listings that include a “dryer” mean a drying rack, not a drying machine. If you do find a place with a dryer, it is likely a washer/dryer combo that washes/dries a tiny load in about 5 hours. You will probably be doing laundry every day. Another reason to stay in one place for at least 3 nights at a time - to give that wash time to dry on the rack.
Personally, I would start in Rome for 5 nights, then Florence for 3 nights, then Venice for 3 nights, then Salzburg for 4 nights (traveling by train between these cities). Rent a car in Salzburg and go to Hallstatt and Berchtesgaden as day trips. Then fly to Paris for 5 nights. Then train to London for 5-6 nights. Fly home from London. This gives you 4 very different cultures, hits lots of iconic places, sprinkles in some smaller places, and gives you 2-5 full days in each place. If this is a summer trip, go immediately after school gets out. We noticed considerable difference in crowds from mid June to early July.