RFS are unhappy with Hodgson's apology and the FA's response, claiming they have not taken into account the offence taken to the use of the word 'monkey' in the wider community.
Hodgson used the term during his half-time team talk in the game against Poland, intending to encourage Chris Smalling to pass the ball more often to Andros Townsend.
He was referring to a joke originating from American space agency NASA during the period they sent monkeys into space - over 50 years ago.
'Monkey' has since become a word synonymous with the racial abuse of black footballers and RFS feel the FA's referring to Hodgson's choice of words as an "innocent remark" falls well short of the required response.
In a letter to the FA chairman Greg Dyke, they say: "We urge you to act upon the recent comments made by the England manager Roy Hodgson to players Andros Townsend and Chris Smalling.
"To announce that the matter is "closed" without any action being taken against the England manager is unacceptable and wholly inconsistent with your policies on equality and diversity.
"For Mr Hodgson or anyone else to make an apology if they fail to understand why the words used were offensive or potentially is not the response of an institution with the resources and profile of the FA. The "innocent remark" made out of ignorance is sadly a common feature of football.
"We...urge the FA to provide mandatory 'race appreciation' training and 'cultural capital and cultural intelligence' training to Roy Hodgson and all football managers in the UK.
"The willingness of those in football to close ranks is reflective of a failure to understand the damage that this type of comment causes whatever the so called "context". The consequences have an impact beyond the realms of football given the position and status of the England manager.
"Roy Hodgson's comments may not have been made with the intention of causing offence as he has explained with the backing of the FA, nonetheless, offence was caused. As noted by Lord Ouseley (Chairman of the anti-racism campaign Kick It Out) in response to Hodgson's comment, where an offence is caused, the FA needs to investigate and report its findings efficiently.
"The failure of the FA to deal with this matter appropriately is explained by its assumption that as no member of the England came forward that there is no victim. The FA ought to be aware that there is a wider BME (black and minority ethnic) community, many of whom found this comment offensive, however unintentional, as having racist overtones.
"This is not a question of political correctness but the simple application of common sense, respect and decency by those in positions of power and authority within the game."
Since submitting their letter to the FA, Race for Sport has issued an unreserved apology for any offence caused to Andros Townsend and his family over a reference to the player. Race for Sport has sent the FA a revised version of the letter - and says the previous comment has been retracted having been submitted in error.