Hodgson's men entered the tournament with high hopes but their new crop of young players failed to deliver, with consecutive 2-1 defeats to Italy and Uruguay ending their prospects of making the last 16 after just six days.
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When told that Italy coach Cesare Prandelli had offered his resignation following their elimination, Hodgson was asked again by reporters about his job but snapped back: "I told you [press] the other day [I won't resign] and I've nothing more to add.
"The FA has asked me to continue. They want me to continue. I'm very happy that they want me to continue.
"I think there is an interesting group of players here to work with and I get no feeling whatsoever that any of them will want me to resign. I have no reason to do so. I have no intention to do so. I don't want to be compared to other people.
"The reason I'm staying on is that I'm not a quitter. I believe in this team and that the FA seriously want me to keep doing this job, as do the players, so therefore I will continue to do it and I'll try and lead them to Euro 2016 and try to get some good results."
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"It would be nice if they don't formally retire because, when players do that, they disappear from England selection," he added.
"[We could] leave it to a dialogue, perhaps, whereby at any moment in time I could speak to one of them and say: 'Look, do you want to play in this game, are you up for it?'. That's how I'd really like to see it."