Question: - I conceded a putt in a matchplay event which my opponent then putted and missed.
Is that counted as an extra stroke?
Answer: - No. Once that putt is conceded the shot is considered to have been holed. (Rule 2-4)
Any concession may not be declined or withdrawn.
However if the act of putting was to give his partner, in a four-ball or better-ball match, assistance then the partner is, in equity, disqualified from the hole. (Decision on the Rules of Golf 2-4/6)
Question: - If a player lifts a ball believing that the opponent had conceded the shot when they had not what should we do?
Answer: - This is a difficult one. If the statement made could have reasonably led the player to believe that the shot was conceded then the ball should be replaced and played without penalty (Decision on the Rules of Golf 2-4/3).
If not then the player incurs a penalty shot for lifting the ball without marking its position (Rule 20-1)
From these rules it shows that it is best to be sure that the shot has been conceded before picking up your ball.
If you concede a shot and do not wish your opponent to play it then pick up their ball and give it to them.