Is it permissible for a trade in which a third party pays a sum of money and obtains a variable profit, while the intermediary takes a fixed percentage of the third party's profits?

The second partner is merely a broker; he does not own the money but rather takes it from others. It is not permissible to set his fee as a percentage of the merchant's profit due to ignorance (of the exact amount) and uncertainty. If the contract is rendered invalid, the broker is only entitled to a fair market wage (ujrat al-mithl), and he has no share in the profit. As for the contract between the merchant and the capital owners, if the merchant acts as a Mudarib (investment manager) who is not liable for the capital or a specific profit, and he shares in the profit while the capital bears the loss without any negligence on his part, then this contract is permissible.