“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]
(..) - A lawyer? - Isn't it obvious? Lady Croft, my pleasure. - Manfred Powell, QC. - Good morning. - I believe you're quite an authority on things Ancient and Mythological. - Well, I travel. No, no. Mr Wilson said you were quite the archaeologist. Please. (..)2001 Lara Croft: Tomb Raider