Burnley captain Ben Mee told fans who flew a "White Lives Matter" banner above the Etihad Stadium at the start of a 5-0 defeat by Manchester City on Monday "to come into the 21st century".
The banner reading "White lives matter Burnley" was seen shortly after all players and match officials had taken a knee in solidarity with the Black Lives Matter movement.
"The aeroplane that went out at the beginning of the game, I am ashamed, I'm embarrassed, that a small number of our fans have decided to put that around the stadium," said Mee.
"It completely missed the point. The group of lads in there are embarrassed to see that and not what we are about at all.
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"It missed the point of the whole thing of what we are trying to achieve. These people need to come into the 21st century and educate themselves as a lot of us do. I'm really upset that happened."
Burnley also issued a statement at half-time saying those responsible for the aircraft would be handed lifetime bans from the club.
"We wish to make it clear that those responsible are not welcome at Turf Moor," Burnley said. "This, in no way, represents what Burnley Football Club stands for and we will work fully with the authorities to identify those responsible and take appropriate action.
"We are fully behind the Premier League's Black Lives Matter initiative and, in line with all other Premier League games undertaken since Project Restart, our players and football staff willingly took the knee at kick-off at Manchester City.
Read | 'Black Lives Matter' to replace names on shirt in Premier League
"We apologise unreservedly to the Premier League, to Manchester City and to all those helping to promote Black Lives Matter."
Black Lives Matter have been printed on the back of shirts rather than players' names for the first 12 games of the Premier League's return.
The gesture comes following the death of black American George Floyd at the hands of police last month, which sparked protests around the world and led to sportspeople showing solidarity with those demonstrations.
Feature image courtesy: AFP / Shaun Botterill