Kennaugh briefly suspended his Twitter account having deleting the messages shortly after sending them. In one of them he told Pooley to “stop being so self centred”.
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In his apology, Kennaugh said he realised he “came across like a total idiot”.
Pooley, who has recently returned to road racing, questioned why a team was not set up to help her win the women’s Giro d’Italia – where she twice finished second – but there was a men’s team set up in an attempt to win the Tour de France.
Kennaugh, 26, questioned why Sky would financially back a project for a race that “absolutely no one in the UK has absolutely no idea about”.
The Manxman swiftly got rid of his tweets, but not before several Twitter users had taken screenshots. Later on Thursday afternoon, the British champion suspended his account on the social media site after a wave of criticism but returned to make his apology.
Kennaugh rode under the British Cycling programme at London 2012, where he won gold in the team pursuit on the track and has been part of Team Sky since it was established in 2010.
Pooley questioned why Sir Dave Brailsford wasn’t being mentioned in allegations of sexism – which mainly revolve around former BC technical director Shane Sutton – as he played a key role in setting up Team Sky.
She told the Guardian: “Why didn’t anyone ask how it could be that a publicly funded body like British Cycling joined together with a privately funded team – Sky – on a mission to get a British winner of the Tour de France within five years?
“Why wasn’t there a similar plan for the women? The women’s Giro d’Italia was the most important race for women but where was the funding for that? I came second twice and no one from British Cycling offered to put together a team to help me win it.”