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  A report into claims of bullying said British Cycling "lacked good governance" at board level.
  Cookson, who led the organisation for 17 years before becoming UCI president, said: "If there was a lacking, I think we all share in that responsibility."
  He added British Cycling had been scrutinised by UK Sport.
  Asked if he was "pinning the blame" on the funding agency, he said: "As I understand it, they were closely involved in how things were managed and monitored.
  "I'm not trying to dodge any of my responsibility. What I'm saying is there were other people with responsibilities as well."
  The Phelps report into the culture of Britain's cycling team criticised former technical director Shane Sutton and UK Sport, and noted a "culture of fear" existed within the team, according to many staff members.
  "I don't accept there was a culture of fear and I spent a lot of time at the National Cycling Centre," said Cookson, who is seeking re-election after joining the UCI in 2013.
  However, he said Sutton should have been moved following recommendations from the King Review in 2012.
  "One of the recommendations was that Shane be moved out of direct coaching with athletes," he said.
  "As far as I understood in the first half of 2013, that was one of those action-plan elements that was implemented and it was after that in the middle of 2013 I became a candidate and then was elected to the UCI presidency."
  Australian Sutton stayed on before resigning last year amid claims of sexism and was later found to have used sexist language towards cyclist Jess Varnish, though he was cleared of eight of the nine charges against him.
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  Cookson defended his record at British Cycling, saying: "I'm not making any excuses for my behaviour. I think we made incredible progress and some great achievements.
  "The idea we had no respect for athlete welfare is wrong. The relationship between athletes and coaches was something we were concerned about from time to time. Action was taken."
  And he disagrees with MP Damian Collins, who has called for board members from the period covered in the investigation to resign.
  "My former colleagues at British Cycling have been committed to their sport over many years. Many of them are good people who have worked very hard," Cookson said.
  UK Anti-Doping is also investigating allegations of wrongdoing against Team Sky and British Cycling, which both deny.
  "The reputational problems that have been around that team in the past few months have not been helpful. I want to wait for the outcome of UK Anti-Doping's enquiries," he said.
  Cookson said cycling had made significant progress in the battle against cheating during his tenure at the UCI.
  "We were previously considered pariahs in the anti-doping world. Now we have one of he best reputations in sport, I believe," he added.
  BBC Radio 5 live sports news correspondent Richard Conway
  Brian Cookson wants to be re-elected later this year for a second term as president of cycling's world governing body, and sees a fresh mandate as his future.
  However, his past is catching up with him.
  After a 17-year period as the president of British Cycling many believe it's under his tenure that the rot set in within the organisation, a fact addressed and criticised in the recent Phelps review.
  Cookson doesn't shirk responsibility for some of the major welfare issues that have occurred within British Cycling. But what is striking is that he refuses to accept wholesale blame.
  In my interview with him, he points the finger firmly at UK Sport, saying one of their officials was embedded within the world-class cycling programme at the Manchester velodrome and was involved in management and monitoring.
  Any "lacking" must be shared, he believes. His message is, in effect: 'Why should the blame fall only on my shoulders?'
  Shane Sutton, according to the 2012 King report, should have been moved away from direct coaching with athletes. We now know he referred to disabled cyclists as "gimps and wobblies" in addition to his role in Jess Varnish's departure from the British team.
  But Cookson points to the fact he moved in 2013 to be president of the UCI and, in his absence, Sutton carried on working with cyclists unimpeded and with serious consequences.
  Again, Cookson's overarching message is clear: nothing to do with me, guv.
target: Brian Cookson, the head of cycling's world governing body, says blame for British Cycling's failings should not rest solely with him.