The defender was fined £90,000 in October 2012 for his offensive tweet in response to the FA's judgement
in the John Terry racism case.
He subsequently deleted the comments and apologised personally to FA chairman David Bernstein.
The incident occurred after Cole's evidence to an independent FA commission - which found Terry guilty of racially insulting Anton Ferdinand during a match
between Chelsea and QPR in October 2011
- was called into question.
Cole will be making his 102nd appearance for England at Wembley on Wednesday having become the seventh player to reach a century of caps,
The Republic visit Wembley for the first time since 1991 as part of the FA's 150th anniversary celebrations, a schedule that also includes a game in Brazil on Sunday.
Hodgson had caused confusion over who would wear the armband by earlier referring to Frank Lampard's regular role as vice-captain.
Cole a fantastic servant - Hodgson
"Frank and I are happy that [Cole] will lead the team out but Frank still remains the captain of the team," said Hodgson.
The FA later clarified Hodgson's comments and insisted to BBC Sport that Cole would be skipper throughout the game.
Lampard said: "The way he has performed, he has been the most consistent left-back in world football. The gesture is very much deserved and I am very happy for him."
marred the last match between England and the Republic of Ireland 18 years ago.
"It's very important," Hodgson said. "It almost goes without saying that you want fans to respect the opposition and I'm hoping that happens, because Wednesday night is such an important part of the FA's celebrations and I hope our fans are going to come and cheer us to victory.
Lampard happy to hand Cole captaincy
"I was very happy to send out the letter to try to make sure that the game is played in the right spirit, as I know it will be on the field.
"I hope the fans, and the Irish fans, show the level of respect that we would expect for a game of this type.
"I'm looking forward to it very much. Not only have the Republic of Ireland got a fantastic record, but they've reached major tournaments and have shown they're a difficult team for any opponent.
"I think it's very fitting that we now play one of the other home nations and we know we're going into the kind of quality game that will help us prepare for our important games in October."
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