Is taking the remaining money after purchasing an item with another's money for less than the agreed upon price considered unlawful (haram)?

It is forbidden to take the remaining amount of money, and it must be returned to its owner. It is not permissible to interpret it as a fee or profit, for the agent is a trustee and is not allowed to take anything without the principal's permission. Agency for a fee requires an agreement on a known fee, and a sale is only valid for what the seller owns and with mutual consent.