Survation carried out the poll for the paper, finding that nearly 20% of Muslims expressed at least some sympathy with those who go to fight in Syria.
The Sun used that as the basis for a front page story saying one in five Muslims were "sympathetic" to the jihadis of IS, seemingly ignoring the fact that there are also more moderate groups fighting in the country.
The story triggered the a record number of complaints to press watchdog Ipso - over 1,000, more than The Sun's infamous opinion piece about migrants by then-columnist Katie Hopkins.
Though The Sun said it stood by its work, Survation released a strongly-worded statement today that said it does not "support or endorse" the paper's reporting.
Survation's chief executive Damian Lyons Lowe and director of research Patrick Brione wrote: "There is a distinction between the work we do and how clients chose to present this work.
"Survation do not support or endorse the way in which this poll’s findings have been interpreted.
"Neither the headline nor the body text of articles published were discussed with or approved by Survation prior to publication."
They added: "Furthermore, Survation categorically objects to the use of any of our findings by any group, as has happened elsewhere on social networks, to incite racial or religious tensions."
They said the best way to interpret their polling results was to compare it with a sample of non-Muslims, praising the "balanced" way Sky News did this when reported on a Survation poll in March that used identical methodology and the same question wording.
This analysis, Lyons Lowe and Brione wrote, shows that sympathy with those fighting in Syria was a "limited, minority" view among Muslims and non-Muslims alike, particularly among young people in both groups.
They wrote that the most "pertinent finding" of Survation's poll was the fall in sympathy for those travelling to Syria, since the poll for Sky News found 28% expressed sympathy for them.
The Sun's normal pollsters, YouGov, reportedly declined to do the latest poll.
A YouGov spokesman told HuffPost UK: "We don’t comment on specific commercial discussions.
"Broadly speaking, in our experience, to survey Britain’s Muslim population, particularly at a period of such heightened sensitivities, requires the kind of time, care, and therefore cost, that is beyond a newspaper’s budget."
Ipsos Mori, another polling company, expressed reservations about the poll.
Its chief executive, Ben Page, tweeted that Muslims were too small a proportion of the British population and polling them online could be "fraught with bias" and polling them by phone, as Survation did in this case, could be "equally hard".
Surveys of Muslims in UK - at 5% of population many online approaches can be fraught with bias...— Ben Page, Ipsos MORI (@benatipsosmori) November 23, 2015
@dylsharpe I know. Tbh sampling Muslim by phone equally hard - esp for likely budget off fast turnaround newspaper poll— Ben Page, Ipsos MORI (@benatipsosmori) November 23, 2015
Dr Shuja Shafi, Secretary General of the Muslim Council of Britain said: “Many Muslims will find this poll hard to believe. The vast majority of the almost three million British Muslims abhor terrorism.
"Poll after poll attest to this, as do the many surveys showing how almost all British Muslims would report someone from the Muslim community to the police, if they knew they were planning an act of violence.”
“Of course, even one person harbouring sympathy for the Daesh death cult is one too many. Terrorism is indeed a problem and many a British Muslim parent are worried whether their children may be lured to go off to in the chaos in Syria.
"But dubious headlines as that printed in The Sun do not help matters. The grand strategy of Daesh is to divide our communities and stoke fear between communities. We should not play their game.”
Speaking before Survation's statement, a Sun spokesman told HuffPost UK the paper was not commenting beyond a strident leader titled 'Face The Facts', published in today's paper.
"Most of our survey, commissioned after the Paris atrocities, probed Muslim attitudes to IS specifically," it said.
"No one agreeing to the statement... was in any doubt which fighters we meant.
"Some on the political Left claim ours was a 'rogue' poll. In fact, the numbers expressing sympathy for jihadists were down on similar survey by the BBC and Sky after the Charlie Hebdo massacre.
"All three polls reveal an undeniable truth. Among British Muslims, a minority - but a substantial one- appear sympathetic to a death cult which is among the most evil in history."
On Twitter, one person said Survation should complain to the Sun and not undertake any polling for it or News UK again.
Suggest a correction
@Survation and announce you won't conduct such surveys in future for News International.— rob (@robfraser111) November 24, 2015