Hibernian captain Liam Craig admits he and his team-mates will be lucky to keep their Easter Road jobs after leading the club to relegation.
But chairman Rod Petrie says he will not walk away from the challenge of rebuilding the broken Leith side.
Terry Butcher's men blew a two-goal lead over Championship runners-up Hamilton in their play-off final second leg before seeing their 15-year stay in the top flight snuffed out after a 4-3 penalty shoot-out defeat.
It was another embarrassing low for a Hibs team who have won just once in their last 15 games, and around 500 fans gathered outside the stadium to blast Butcher and the board after the game.
Craig said: "Th e fans have got every reason to want to get rid of the manager, the players, everyone.
"We have been an embarrassment. We are in, in the morning for meetings about what lies ahead. That was the plan anyway and I can't see that changing.
"And I think if you are here next season then you will be very, very lucky."
But in a statement posted on the club's website, Petrie refused to quit after apologising for a "dismal season".
He said: " We recognise that change is needed.
"Some supporters have suggested that I should resign as chairman. I take responsibility for the position that the club finds itself in.
"However, it would be wrong for me to abdicate that responsibility at this time and walk away and it would be wrong for the club.
"Having already announced a few weeks ago that I am handing over my executive responsibilities, it is in the best interests of the club for me complete that transition to our new chief executive and to oversee the other far reaching changes at the club as non-executive chairman.
"Leeann Dempster has already been appointed as chief executive to lead the club forward. Leeann achieved success at Motherwell and will work closely with Terry Butcher as the club begins the task of assembling a competitive squad for next season.
"Leeann's arrival is the first of a number of planned changes in the coming weeks and months ahead at Hibernian."
Jason Cummings' double at New Douglas Park on Wednesday had put the Edinburgh men on the verge of safety.
But Jason Scotland's 12th-minute strike and a last-gasp leveller from Tony Andreu dramatically swung the game back in Accies' favour.
With no more goals in extra time, the match went to penalties but Hibs blinked first when Kevin Thomson saw his opening effort saved by Accies goalkeeper Kevin Cuthbert.
And Hamilton's number one put the seal on their promotion party when he blocked Cummings' tired effort.
Cuthbert made his debut for former club St Johnstone in 2001 - the season they were relegated from the Premiership - and has spent the last 13 years trying to fight his way back.
"All the years of efforts and knock-backs are now worth it," he said. "I was released by Morton three years ago but always said that I wasn't going to let that defeat me.
"I have always had faith in my own ability and that has never wavered. But this is an amazing feeling and worth all the hardships I've had to handle in my career."