“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]
(..) would come look at my room and say, ''Would you look at this filth?'' Now, l've already been in the room five hours. She wants me to look at it. My mother was an authority on pigsties. ''This is the worst-looking pigsty l have ever seen in my life! ''And l want it cleaned up right now. (..)1983 Bill Cosby: Himself