“Who gave him the 'authority' to do this?” “He didn't do it 'on his own authority' [=he didn't do it on his own]; he was acting 'under the authority of' the company president.” [=the company president gave him the power to do it]
No one 'in authority' objected to the plan. = No one 'in a position of authority' objected to the plan. [=no one with official power to make important decisions objected to the plan]
This office has 'authority' over personnel matters. [=this office has the power to make decisions about personnel matters]
(..) This is hardly the time or the place. Then we'll state another time and place. Tomorrow at in the prefect's office. With mademoiselle. Capt. Renault, I am under your authority. Is it your order we come to your office? Let us say it is my request. That is a much more pleasant word. Very well. (..)1942 Casablanca