Conversation so far:
What is Rogue's relationship to Mystique in X-Men?
Mystique becomes the adoptive mother of the four-year-old girl Rogue.
What happened to Rogue's parents in X-men?
Rogue had run away from her home in rural Caldecott County, Mississippi.
What are Rogue's powers in X-men?
Her mutant power is the ability to absorb the memories, personality, and skills or powers of whomever she touches.
Does Rogue join Mystique in brotherhood?
Rogue is trained by Mystique and eventually joins the Brotherhood of Evil Mutants.
Does Rogue turn on Mystique?
Rogue tells Mystique that she joined the X-Men because Professor X, as the world's most powerful telepath, is her best hope of healing for her fragmented psyche.
Is Rogue successful at freeing the other members of Brotherhood?
The Avengers ultimately defeat the Brotherhood of Evil Mutants, capturing all of them except Rogue and Mystique.
Do Rogue and Mystique go into hiding from the Avengers?

Question: Where does Rogue go when Mystique give up guardianship?
Self-contained rewrite: Where does Rogue go after Mystique gives up guardianship?