Using standard ECC - Role of the Supervisor documentation it reasonable to assume Supervisor duties will include supporting the PM in reviewing /commenting/ accepting proposed materials / working processes…
In short - No!
The role of the Supervisor is pretty much solely to monitor that the works are carried out in accordance with the Works Information, identify defects and get involved with tests and inspections. The only instruction the Supervisor can specifically give under the contract is to uncover works if they believe a defect may exist.
On a practical level they may assist the PM more, and may even have delegated powers from the Project Manager, but otherwise that is the limit of their responsibility under the contract.