Clyde Valley Flute Band from Larne, wore the emblem with the letter ‘F’ on its shirts during Saturday’s Apprentice Boys parade.
Tony Doherty, the trust’s chairman, described the events as a setback.
The Apprentice Boys again said there was no prior agreement on symbols.
Mr Doherty also said the Apprentice Boys had yet to respond to the trust’s request to meet, but added he expected one soon.
Apprentice Boys governor Graeme Stenhouse told BBC Radio Foyle the association is open to meeting “any groups in the city who wish to discuss the incidents and issues from last week”.
Thirteen people were shot dead when members of the Army’s Parachute Regiment opened fire on civil rights demonstrators on Bloody Sunday in Derry in 1972.
A 14th person later died of their injuries.
An ex-paratrooper, known as Soldier F, is facing prosecution at a Derry court next month for two murders and four attempted murders.
On Tuesday, the Apprentice Boys said they recognised the potential upset caused to nationalists.
Mr Doherty, whose father Patrick was among those killed, said the trust had given “a guarded welcome” to the statement.
“In our view there is much more to be done to restore relationships in our shared city,” he said.
“We value relationships and wish to see our city prosper and thrive.
“The events of last Saturday are clearly a setback.
“We must all play our part in ensuring that the full gravity of the situation is acknowledged and understood, that the full facts of how it came about be established, and to receive full assurances that the matter will be dealt with properly allowing for no repeat ever.”
He said the trust is due to meet the PSNI on Friday to discuss its concerns.
“We also fully expect to receive a response from the Apprentice Boys soon,” he added.
Mr Stenhouse told BBC Radio Foyle the association is hoping to keep the “good dialogue that’s been going on for the last 20 years with different groups in the nationalist community”.
“We will be hoping to speak to them at the earliest opportunity to discuss any issues they have.”
Mr Stenhouse again said there was no prior agreement with police or other parties banning any emblems and said senior officials from the association will meet with the Clyde Valley band later on Thursday.On Wednesday, four of the 145 bands who marched on Saturday pulled their support for the Apprentice Boys of Derry in light of their statement condemning the Clyde Valley Flute Band’s actions.
Rathcoole Protestant Boys said they would not participate in another Apprentice Boys of Derry parade again, while Cloughfern Young Conquerors and Pride of Ballybeen have also decided to terminate contacts with the Derry organisation.
The Pride of Greenisland said it too would be unable to attend any future Apprentice Boys of Derry parades in the city.
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