Equality group Kick It Out have called for an investigation into comments made by Roy Hodgson at half time of England's 2-0 win over Poland.
Hodgson, 66, told a joke about an astronaut and a space monkey to illustrate the need for Chris Smalling to get the ball to Andros Townsend in the second half.
In a statement, the group said: "Kick It Out, football's equality and inclusion campaign, recognises and shares the concerns of the parties who felt mindful to bring the comments into the public domain.
"The matter has been raised by the Chair, Lord Herman Ouseley, directly with The Football Association (FA), who acknowledges the apology made by Roy Hodgson, and now seeks an investigation to ascertain the full facts and ensure a similar situation does not arise again."
I am quite happy that we have shown to you that we are not scared to spend money when we think the players have the quality and we have the funds available to do it.
I believe we have a huge advantage in that our club is well respected all over the world because we have values, because we are not artificial, because we are based on tradition, we are as well forward thinking and we give a chance to young people who deserve it.
That is for me something that is very important to preserve.
Arsenal manager Arsene Wenger says the club is ready to spend again on buying more world-class talent, like Mesut Ozil. Wenger spoke as Arsenal held their annual general meeting, having smashed their transfer record with the £43 million deadline day signing.
However, Arsene Wenger told shareholders the club was not about to abandon its long-standing principle of youth development.
He said: "The development of our club has to be based on the intelligence of our eye. That means players who are unknown and who can become big players here, through our connections, the quality of our scouting, through the quality of what we have seen in them and how they could develop.
"Then, part of our development and ambition is to scout and buy recognised world class players. This season for example we have bought Ozil. You don't need any scout to buy Ozil. You just need money.
"I am quite happy that we have shown to you that we are not scared to spend money when we think the players have the quality and we have the funds available to do it."