The former Girls Aloud singer confirmed on Monday that she would be returning as a judge on the upcoming series of The X Factor, along with Cowell, after a four year break.However their friendship had suffered a blow in 2011 when they both headed to the US to launch the show in America, only for Cowell to sack Cole after just two days on the job.But despite falling out over the incident, Cole insists all is forgiven and the pair now have a "great relationship".Cole's return to the UK show was announced by Cowell with a message posted online, which said: "It's official. Cheryl is back. Most importantly she is a brilliant judge of talent."The singer said Cowell had begged her to return to the show for years, joking: "It's not nice to see a grown man cry."Speaking to Daybreak on Wednesday, new father Cowell admitted he got it wrong by firing Cole and said it took "2,000 calls" to persuade her to forgive him and come back.When I decided I wanted to come back, I also decided I had to have Cheryl back on the show.Simon CowellHe said: "Of course I was wrong and I don't mind admitting it. If I could turn the clock back I'd do it differently, but I can't."Cowell said he believes the pair can pick up where they left off as he considers her a friend, saying: "I trust her and she is loyal."He added: "I have done this a long time, you have fallouts along the way. I held my hands up and said I had made a mistake and I was sorry.And it seems the battle lines have already been drawn between the pair over who will join them on this year's judging panel.Cole said she would like another woman to join her on the panel, while Cowell said he would prefer to keep show veteran Louis Walsh.But Cole revealed that Cowell had given her the final say on who to recruit, adding: "There are people I'd love to join the panel. Definitely a strong woman. I'm kind of hoping I don't have to deal with three guys because otherwise I'll have no hair - I'll be pulling it out all the time."Cowell said the two of them would discuss who to bring in, saying: "When Cheryl came back we decided we'd agree together. I'm not going to put someone beside her who she hates."A spokesman for Walsh said: "ITV and Simon have given assurances he will be back."The X Factor returns to UTV in August.