Did you actually do this yet, or only planning for it? If planning, you really should see an attorney who specializes in estate planning to make sure your needs are met without screwing up and owing taxes.
If it already happened, then the math goes something like this.
1. If your aunt bought the home with your parents, her starting basis is 1/3 the price they paid. (If she was put on the deed in some other way, this can be even more complicated, and more information is required.)
2. Then add 1/3 the cost of any permanent improvements made to the home between the purchase and the death of your father.
3. Then the aunt inherits 1/6 the home (half your father's 1/3 share) at its fair market value on the date of his death.
4. Then add 1/2 the cost of any permanent improvements made while your mom and aunt shared ownership after your father's death.
5. Then, your aunt probably gifted 1/3 of her interest to each other person; your mother, your sister and you. Or, your aunt gifted 1/2 her interest to you and your sister. If the deed does not specify exactly how the interests was split, it will be governed by state law. Regardless, she also gifted you the same share of her cost basis.
It also matters very much if your aunt transferred the home to you in fee simple, or as joint tenants with right of survivorship with your mother. Because if your mother has a right of survivorship (can live in the home until she dies) then you don't really own anything now and will inherit a stepped-up basis equal to the home's fair market value on the day she dies (half each to you and your sister). But if you all sell it, then you each own capital gains tax based on 1/3 the sales price minus the dollar amount of your basis.
