Isobel is Ursula's aunt. She is a few years younger than her brother, Hugh (who's Ursula's father), and one of the first major events of the book is when Izzie has a baby at sixteen. In most of the timelines, the child is adopted by a German couple, except in one timeline in which she gives birth at Fox Corner. (The child drowns a few years later.)
Later, she writes—under the pseudonym of Delphine Fox—a Bridget Jones-esque column called Adventures of a Modern Spinster, which she manages to spin (ha) into a book series about her fictional nephew, Augustus. This allows Izzie to do something most women at the time either don't or can't do: make a comfortable living for herself financially. It's a good thing for Izzie, who has a love for big houses and fast cars.
Ursula is able to go to her aunt Izzie for things she can't get from her mother. As a result, the two form a special bond. At one point, Izzie tells Ursula "you remind me of a little me" and Ursula wonders, "Was that a good thing?" (20.45). Although they're close, their relationship isn't free of bumps. In the timeline where Ursula gets pregnant, she resents Izzie for taking her to get an abortion, especially since Ursula believes she's going to give the baby up for adoption. No one likes a surprise surgery.
Izzie and Ursula's mother, Sylvie, do not get along. The women have completely different values, with Sylvie being more traditional and Izzie being more modern, or, as Sylvie would probably assess, loose and promiscuous.
Izzie is a character who remains pretty consistent throughout most of the timelines. At one point, Hugh observes that "Izzie is Izzie" (20.109), and Ursula says, "Izzie, apparently, had become herself a long time ago" (20.109). And that's the best way to describe Izzie: unapologetically herself, even when she's having her car repossessed or trying to adopt another woman's children. Izzie is Izzie.