The group - which created a missionary bishop to cater for disaffected Anglicans in Scotland, England and across Europe who are unhappy with the church's teaching on issues such as sexuality - made comments following Most Rev Justin Welby's criticism of the body at the recent Primates' Meeting on Friday.
Archbishop Peter Jensen, GAFCON's general secretary told Premier that Archbishop Justin was appointing blame to the wrong people.
Speaking about the teaching and practice surrounding sexuality, Archbishop Peter said: "The Communique seems deliberately to step around the real problem and then offer blame to the people who are really in a sense the victims.
"By putting so much emphasis on blaming people who are so-called crossing boundaries, but avoiding the real issue which is the flagrant breach of the teaching of the word of God by a number of the Anglican churches around the world."
At a press conference for the Communion's Primates' Meeting, Archbishop Justin said that those who have started churches without the permission of the bishop in the diocese or province "breaks trust and weakens the communion".
He added: "The problem is where... one group subverts the efforts of another church group.
"Jesus created the Church to be one so that the world might know that he came from God... and that's really crucial, and failure to do that is a cause for repentance."
Archbishop Peter questioned the validity of the meeting itself.
He told Premier: "This was not a meeting of the primates. Three of the most significant primates [Nigeria, Uganda and Rwanda] did not attend on principle.
"When you get that situation and then the reason for them not being there is studiously avoided - the whole thing becomes in a sense a little farcical."
GAFCON has repeatedly expressed fears of perceived leniency on issues of sexuality but told Premier it won't leave the worldwide Anglican family, as they "love the Communion".
Archbishop Peter went on to say: "We are going to have our first great meeting in Jerusalem next year and the bonds of fellowship between the GAFCON churches and GAFCON membership is continuing.
"We are not leaving the Anglican Communion - we are seeking the reformation and the renewal of the Communion."
