The novel begins with the Madden sisters and their childhood friend in Clevedon. After various travails, the adult Alice and Virginia Madden move to London and renew their friendship with Rhoda, an unmarried bluestocking. She is living with the also unmarried Mary Barfoot, and together they run an establishment teaching secretarial skills to young middle-class women remaindered in the marriage equation.
Monica Madden, the youngest and prettiest sister, is living-in above a shop in London. She is, in modern parlance, "stalked" by a middle-aged bachelor Edmund Widdowson, and he eventually brow-beats her into marriage. His ardent love turns into jealous obsession suffocating Monica's life.
Meanwhile Mary Barfoot's rakish cousin Everard decides to court Rhoda initially as a challenge to her avowed dislike of love and marriage, but he later falls in love with her for her intellectual independence, which he finds preferable to the average uneducated woman's inanity. Despite being virulently anti-marriage, she decides to indulge him with a view to turning down any marriage proposal to show her solidarity with her "odd women". Ironically, she in turn falls for him.
Married Monica meets Bevis, a young, middle-class man who pursues her and represents for her the romantic ideal from popular novels. Crucially, Bevis lives in the same building as Everard Barfoot. Monica, determined to elope with Bevis, goes there. Unbeknownst to her, her husband has hired a detective to follow her. She hears someone follow her up the stairs and, to appear innocent, she knocks on Barfoot's door. This is reported back to Widdowson, and he feels his suspicion has been justified and informs Mary Barfoot of her cousin's blackguardly ways.
Rhoda, on the other hand, is on a holiday in Northumberland, and Everard goes to see her there. He woos her and at first suggests they enter a free-union (i.e. live together out of wedlock), which would appear to be consistent with her principles. However, she gives him a conventional "womanly" response and agrees to be with him only in a legal union; Barfoot, somewhat disappointed in her surprising conventionality, proposes marriage, which she accepts. She then receives a letter from Mary telling of Everard's supposed affair with Monica. Rhoda then breaks off the engagement, after Everard proudly refuses to give an explanation but insists he is innocent. After Widdowson confronts Monica over her infidelity, she leaves him but lives at his expense and even moves, together with her sisters, to his rented house in Clevedon. Virginia has become an alcoholic (her way of dealing with being an 'odd woman'). Monica is pregnant by her husband, but her pride will not let her reunite with him. To salve her conscience, she visits Rhoda and shows her a love letter from Beavis and also exonerates Everard over the alleged affair. Then, months after they last saw each other, Everard visits Rhoda, asks her if she still believes him to be guilty, and repeats his offer of marriage. Even though Rhoda assures him that she believes him innocent, she refuses his proposal, intimating that in his professions of love he was "not quite serious," but was partially testing her principles. It is too late for them to reunite. Barfoot soon gets married to a conventionally educated young woman. Monica gives birth to a girl, then dies soon after. The novel ends with Rhoda holding the baby, crying and murmuring, "Poor little child!"

Question: What does Everard ask of Rhoda, which she refuses, when visiting her in Northumberland?
Answer: To live together.