Is it permissible for my relative to take a commission from the buyer, given that the selling company is aware of the sale of the property but not of the commission, and that his primary job is project supervision, and that he did not reduce the amount the seller would receive?

The project supervisor is an employee acting as an agent for the company. It is not permissible for him to accept gifts given to him because of his position unless he has the company's permission, for the Prophet, peace and blessings be upon him, said: "Gifts to workers are unlawful gain." Furthermore, a buyer's gift to the agent is considered a right belonging to the principal (the company) and an increase in the price. Therefore, your relative must inform his company about this commission. If they permit him to keep it, then so be it; otherwise, he must return it to the company.