Go Blue Plaque hunting.
See how many coal chute covers you can accumulate.
See how many ancient musical instruments you can find.
Take in a concert at Kenwood House.
Swim in a pond on Hampstead Heath.
Go on a short train ride to Berkhamsted and walk around a slighted castle's ruins and then walk the Grand Union Canal.
Autumn is a bit vague. If the equivalent of today's date, take in the uniquely London historical event of the Lord Mayor's Show.
Go to Brands Hatch or Silverstone for a track day, or take in a BTCC race day.
Take suitable precautions and check the tide tables then go mudlarking.
Go to Bethnal Green tube station and find the very sad memorial.
Go to Cable Street and discover what happened there a few decades ago.
Find World War II bomb damage - start on South Audley Street.
Find a (or more) pillbox awaiting nazi invasion.
Go on a Circle Line Pub Crawl.