@lorisparkle
It is really useful to see what comments others have made. Thank you.
I clarified the position relating to people holding the protected characteristic of Gender Reassignment and pointed out that it doesn't mean that people with the pc can access the facilities and services of the opposite sex.
The comparator when considering discrimination against a transwoman (a male with the pc of Gender Reassignment) would be another male without the pc of GR. If they exclude one group of males from a female only group for eg. it is unlawful to include another group of males.
This is one of the most common misconceptions that Stonewall have lied about about the EA 2010. The law is very clear that it is lawful to exclude ALL males from certain spaces and services as long as it is proportionate and a legitimate aim. A likely proportionate aim would be to have a woman only cycling group because it has been found that women are more likely to participate.
I pointed out the issue of exclusion of certain groups of women who are either not willing or able to share groups or facilities with males. The inclusion of a male person in such a group effectively excludes all the women from that group or service.
GDPR is another consideration they need to take seriously. Presumably the 'gender marker' they hold for members is held on a database somewhere. Gender is not on the list of permitted data that an organisation can legally hold so they are likely to be in breach of GDPR and maybe liable for a large fine.
I suggested they ditch their legal advice and get someone who actually can read the EA 2010. 