What actions can a son undertake to atone for his deceased father's error of taking something from an administration that cannot be returned, such as giving charity or fasting?

The father should have returned what he took to the administration. Since he did not, the value of what he took must be taken from his estate and returned to its rightful owners. If that is not possible, such as if they do not accept it and it is public funds, then it should be spent on charitable causes. If the reason for the impossibility of return is its consumption or damage, then an equivalent amount from the estate must be returned. However, if the deceased did not leave any wealth, the heirs are not obligated to do anything, though it is recommended for them to do so as a filial act towards their father, along with frequent seeking of forgiveness and supplication for him.