Buttler has played for England.
Buttler will miss the final Test because of a finger injury.
The man will now fly home after being struck while keeping.
Sam Billings will make his Test debut for England in the day-night match in Hobart on Friday.
In order to join the England squad, he had to drive from the Gold Coast to Sydney.
England have doubts over all-rounder Ben Stokes, who has a side strain, and Bairstow, who was hit on the thumb while making a century.
The three men were the only ones to bat on Sunday as England held on to keep the series going.
Bairstow and Buttler were injured in the first and second days of the match, and Pope took the gloves in the second.
Buttler faced 38 balls for 11 runs during the second-innings rearguard, despite being out for a duck in the first.
Buttler had suffered a serious injury, according to Joe Root.
"For him to front up from the moment he took that knock shows how much he cares, how much it means to play in this team and play for England," said Root.
While bowling in Australia's first-innings, he strained his side and played no further part with the ball, though he did make a half-century with the bat.
Bairstow made a ton of money in the first and second days after being hit by Pat Cummins.
A number of players were hurting and still putting in a lot.
There was a bit of pain relief that was needed. It's more than just that and also the psychological element. The way they did it shows a lot of character.
