The Famous Five Collection 1: Books 1-3 by Enid Blyton

The Famous Five
The Famous Five are a group of children who have the sort of adventures most kids dream about, in a world where ginger beer flows and ham rolls are a staple diet. Julian and Anne get together with their cousin George in the first adventure, Five On A Treasure Island.

George is actually a girl who wants so desperately to be a boy she crops her hair and struts about doing boy things. She hates it when people call her by her correct name, Georgina. She has a dog called Timmy—oh yes, and an island. Most kids just have a dog, but George's parents own Kirrin Island and let her run around on it as if it were her play-thing.