It has been confirmed that neither Anelka nor the FA will appeal against the five-match ban imposed by an independent regulatory commission for the player making an gesture associated with anti-Semitism.
Kick It Out claimed it was "beyond feasibility" that Anelka did not know the 'quenelle' salute had anti-Semitic connotations and said people would be perplexed the FA is not appealing against the leniency of the sentence. It also called on the FA to bring its minimum punishment into line with UEFA's 10-game ban even though the governing body only brought in its five-match rule last year.
Anelka's club West Brom have also come in for criticism with Kick It Out saying the Baggies should have apologised for the offence caused by the striker's goal celebration against West Ham on December 28.
Anelka had told the commission he made the gesture in support of the French comedian Dieudonne M'bala M'bala, who has been convicted seven times of anti-Semitic crimes, but that he was unaware of any anti-Semitic associations.
A statement from the group read: "Kick It Out finds it difficult to reconcile the sanctions imposed and conclusions reached by the commission.
"There are many ambiguities that are not easily reconcilable with the final outcome. For instance, Anelka has attended one of Dieudonne's live performances and admits to being a friend of his, and to claim that he does not understand the anti-Semitic connotations of the quenelle is beyond feasibility.
"The FA provided a hard body of evidence, complemented by in-depth research into the quenelle by an appointed expert, to justify a stronger ban than its five-match minimum.
"People will be perplexed as to why it has not been challenged considering they were clearly seeking a sanction which was quite different.
"Kick It Out urges the FA to review its anti-discrimination regulations and increase its minimum ban for players found guilty of discriminatory acts to 10 matches, in line with UEFA, so as to provide a much more meaningful deterrent to deal with potential offences."
West Brom have also come in for criticism from Kick It Out.
The statement added: "It was clear from the outset of this incident that Anelka's conduct caused great offence to many people, both within the Jewish community and the wider public.
"His employer West Bromwich Albion should have exercised its leadership as a community institution to apologise for the offence caused and deal with the matter in accordance with its disciplinary procedures."
Both Anelka and the FA had seven days in which to appeal the sanction but the FA's director of governance Darren Bailey confirmed no such action would be launched.
Bailey said: "We do not consider there is a real prospect of successfully appealing to extend the sanction imposed.
"The grounds of appeal available to us are limited to legal challenges or to circumstances in which the sanction imposed is 'so unduly lenient as to be unreasonable'. That is a high test."
Bailey confirmed the FA had pressed the commission to impose a harsher sanction than five matches.
He added: "It is not so far outside the range of sanctions that would have been appropriate in this case to be properly described as unreasonable."
Anelka will therefore sit out the upcoming trips to Swansea, Hull and Norwich as well as the home games against Cardiff and Tottenham.
He will be available to return against his former club Manchester City on April 21, a game which is followed five days later by the return fixture against West Ham.