A number of members from the well-known supporter group posed outside Parkhead with a banner reading: “Save the ten – Time to go Neil”.
They also released a statement on social media to accompany the stunt.
An excerpt read: "Neil Lennon and many of the current Celtic squad have delivered unprecedented success for our club and deserve enormous credit for their achievements.
"However, past accomplishments are not enough to excuse the level of performance and lack of desire we have endured throughout this momentous season.”
Celtic are bidding to win an unprecedented 10th consecutive league title this season, but are currently 11 points behind Old Firm rivals Rangers at the top of the Scottish Premiership.
The Hoops have won just one of their last four league outings, and have also struggled in the Europa League, taking one point from their opening three matches on the continent this term.
But despite his former side’s relatively poor run of form, Sutton has spoken out in defence of Lennon, and has heavily criticised The Green Brigade for their attempt to oust him.
Speaking to the Record Celtic podcast, he said: “I thought that the stuff with the banner outside the stadium was just a bunch of idiots, I really mean that because what good can it do?
“What positive does it do. We all love the game, we all have different opinions about things. I’ve got a lot of Celtic supporting friends, they have texted me saying that ‘Neil Lennon has been a great servant to the club, great player, he’s had great success as a manager and it is time for him to move on’ in their opinion.
“And you know what, I understand that and I understand the frustration with a lot of fans out there at the moment. They’ve had it so good for a long, long time but you know the stuff, the banner outside the ground this morning, there was a sense of entitlement and just spoiled, I don’t want to swear but I just don’t get that, I don’t get the upside of that. There wasn’t one. It’s totally disrespectful to a guy who, the manager who has served the club with aplomb.
“I’ve had an association with the club for a long, long time. I actually think what happened outside the ground is shameful, unnecessary, unwarranted. I feel sad for Neil with this as much as anything. I actually do, what he has actually given to the club. You know he has toiled for years and worked hard as a player, given his absolute all, his heart and soul to Celtic Football Club and then for people to treat him like that.
“If you want him to go then fine, that’s an opinion but I think that there is a way to do it and to do it with a banner outside the stadium as they did this morning, I thought was despicable.”