“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]
(..) No, Natasha, she...she likes to flirt. I've seen her flirt, up close. This ain't that. Look... As maybe the world's leading authority on "waiting too long"...don't. You both deserve a win. What do you mean, "up close?" - But, it's a trick! - No, no, it's much more than that. (..)2015 Avengers: Age of Ultron