I think that arya is trying to make little finger think that she is falling for his plan. Think about it , arya does not want to be a lady or just in one place, I think she likes traveling and Sansa fought them killing Ned.and then she hands the blade that was meant to kill BRAN. who would know the truth about littlefinger???? BRAN!!! So Sansa had to send breanne away so littlefinger could make his move!