The Equality and Human Rights Commission (EHRC) is a public body, funded by the British government, which is responsible for promoting and enforcing equality and non-discrimination laws in England, Wales and Scotland. Possibly its most high profile activity in recent years was its inquiry into allegations of antisemitism within the British Labour Party.
However, there have also been various different allegations that the EHRC is itself engaged in unequal treatment of some groups. The merit of these allegations not withstanding, which body would be responsible for conducting an inquiry into the EHRC, it if were suspected of discrimination against one or more groups at an organizational level?
Possible answers include: the police, but if there is no suspicion of criminality this would likely be outside their remit; the EHRC itself, which has obvious conflict of interest issues; the Government Equalities Office, which is part of the Cabinet Office, and is the EHRC's "sponsor", but again there may be a conflict of interest, and doubts over whether they are equipped to undertake such an investigation; the Office of the United Nations High Commissioner for Human Rights, this appears to fall under their mandate but it's unclear whether they are the "official" overseer of the EHRC according to the UK government or if they would unilaterally undertake an investigation.
I suppose it is possible that, at its foundation, it didn't occur to anybody that the EHRC itself might need to be investigated at some point. Nevertheless, it seems likely that there is an existing organization that could conduct such an investigation if necessary, I just don't know what it is.