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All three England players missed penalties in the 3-2 shootout loss and were targeted on social media after the game.
The FA said it was “appalled” by the abuse and the Metropolitan Police is investigating.
In a statement, the FA said: “The FA strongly condemns all forms of discrimination and is appalled by the online racism that has been aimed at some of our England players on social media.
"We could not be clearer that anyone behind such disgusting behaviour is not welcome in following the team.
“We will do all we can to support the players affected while urging the toughest punishments possible for anyone responsible.
"We will continue to do everything we can to stamp discrimination out of the game, but we implore government to act quickly and bring in the appropriate legislation so this abuse has real life consequences.
"Social media companies need to step up and take accountability and action to ban abusers from their platforms, gather evidence that can lead to prosecution and support making their platforms free from this type of abhorrent abuse."
The Metropolitan Police said: “We are aware of a number of offensive and racist social media comments being directed towards footballers following the Euro 2020 final.
“This abuse is totally unacceptable, it will not be tolerated and it will be investigated.”
Prime Minister Boris Johnson condemned the abuse.
Mr Johnson said: “This England team deserve to be lauded as heroes, not racially abused on social media. Those responsible for this appalling abuse should be ashamed of themselves.”
The Duke Cambridge released a statement on Monday morning.
“I am sickened by the racist abuse aimed at England players after last night’s match,” it read.
“It is totally unacceptable that players have to endure this abhorrent behaviour.It must stop now and all those involved should be held accountable. W”