Previously, Iraqi deportees from Britain have been flown into the Kurdish region of northern Iraq, which has not seen the level of violence experienced in other parts of the country.
The Ifir said some of the asylum-seekers who were left in Iraq had been forcibly removed from the aeroplane.
The organisation quoted one of the people who disembarked in Baghdad as saying: "We couldn't move on the flight there were so many security. Even the food was expired.
"I'm too scared to go to where I used to live. Everything they told us is a lie"
One refugee quotes by the International Federation of Iraqi Refugees
"They forced 10 of us to get off in Baghdad. They said the British embassy would help us, but they just gave us $100 and left us.
"I'm too scared to go to where I used to live. Everything they told us is a lie."
Ifir said those returned to the UK were now back in Brook House detention centre in London.
Ali al-Dabbagh, the Iraqi government spokesman, told Al Jazeera that eight refugees had been accepted and 36 were returned to the UK because they did not have "proof that they are Iraqis".
"We do need to be notified by the UK authorities. We do need to check them [refugees]; to check their identity. This is the normal procedure, which is being followed. Unfortunately, the UK authorities just put them on the plane and sent them to Baghdad," said al-Dabbagh.
"They don't have any proof that they are Iraqis ... The UK didn't follow the normal procedure. This is not the right way. We're against forcing them to be deported.
"As you know, many refugees or asylum seekers are claiming that they are Iraqis whether they are in the UK or Europe in order to get an easy status of asylum."
In a statement on Friday, Lin Homer, the chief executive of the UK border agency, said: "We are establishing a new route to southern Iraq and have successfully returned 10 Iraqis to the Baghdad area. This is an important first step for us.
"We are working closely with the Iraq government to iron out the issues which lead to some of the returnees being sent back, and expect to carry out another flight in the future.
"Having an enforced route for returns is an important part of our overall approach; however, the government prefers the majority of returnees to leave voluntarily.
"In the past three years, more than 2,500 people have chosen to return to Iraq under the Assisted Voluntary Return Programme and we expect that to continue."
According to Britain's interior ministry, 632 people were deported to northern Iraq between 2005 and 2008.