The charity Rhythmix was worried that the existence of a band with exactly the same name could affect their work and according to the BBC accused the X Factor show of not having considered "any of the moral implications of their actions" when selecting the band name. A quick check of the UK Trade Mark register reveals that the charity's UK trade mark registration for Rhythmix covers "education" in class 41. However, the charity's website reveals that Rhythmix is "one of the UK’s leading music charities, delivering a large range of music making opportunities for young people". Also see the IPKat's earlier research on here and a little thank you to Andy J.
As it turns out, the girl band has already agreed to change its name and will introduce its new name "in due course". At least all of this has provided the band, the TV show as well as the charity with some free publicity, Merpel muses.