In a way that is so. Lady Arabella does manage to drive Mary to go to Boxall. But some of the people disapprove, and Dr. Thorne is infuriated with Lady Arabella. Lady Arabella also doesn't want Frank to marry Mary because Mary is illegitimate. The letters between Augusta and Lady Amelia revolve around the problem of Mr Gazebee's rank, and what's more Augusta obeys Lady Amelia because Lady Amelia is so much higher in rank.
Money brings rank and rank brings money.
Now one of the ways Lady Amelia dominates Augusta is by referring to rank and duty. We don't see Lady Amelia have many duties except to snub other people, though she argues Augusta had the duty not to marry below her rank. This apparently didn't apply later when Gazebee made money and himself rose in rank. Augusta on the other hand really falls by not marrying. She can get no money unless she marries because her father, Lord De Courcy is really strained too.