What is the ruling on paying a sum of money on behalf of someone to a company to settle their debt with it, in exchange for that person repaying me the amount paid, with an increase?

The issue of "discount and accelerate" (waḍaʿ wa taʿajjal), meaning to reduce a debt in exchange for accelerated payment, has been deemed impermissible by the majority of scholars and the four imams because it is usury (riba). However, permissibility has been narrated from Ibn Abbas, and this was also stated by Ahmad in a narration chosen by Ibn Taymiyyah and Ibn al-Qayyim. The Islamic Fiqh Academy has adopted this view, provided that it is not based on a prior agreement.

As for lending an amount to be paid to a company, on condition that the borrower repays more than that amount in installments, this is explicit usury (riba).