Could this be "Hermione's Circumstances" by PyroAndrea? I haven't had a chance to read through it, but chapter 12 has Hermione dealing with a rude female cousin named Samantha (who enters the scene physically assaulting Hermione in an attempt to learn how Hermione spilled a milk-shake on her without touching it) and later in the chapter, Harry, Ron, and Hermione go roller-skating.
Shortly after their discussion, Hermione and Ron headed back up to there separate rooms to sleep. Morning came quickly and with it came Hermione's family.
CRUNCH! Hermione's eyes burst open with full awareness as a searing pain came from her hand. She looked up, only to find Samantha standing above her.
After breakfast had been consumed, Mr. Granger made an announcement. "Today, we're going to do something fun! I know how bored the children probably are, so I was thinking we could spend the entire day doing things with them! First we're going to go roller skating, and then tonight we're going to this little karaoke place for dinner!" Mr. Granger looked smiling around at all the faces that in general didn't seem too excited.
Reading through the first few chapters, the central conceit is that Hermione is an adopted child, the daughter of her current father's younger brother, a ne'er-do-well who abandoned her and saddled her with his reputation. Her overwork is the result of trying to overcome this background.
Two things which do not match are that Harry shows up on Hermione's doorstep rather than being picked up, and he has Ron with him. A third thing, which I am finding as I read further, is that it looks like it's more of a Hermione/Ron romance. I suspect this is not your story, but it is close enough that it might help someone else in the future.