If you want to / have to maintain organisation-only access to the group, you won’t be able to use the groups.google.com UI to do this. Instead, you can add service accounts to an organisation-only group via the GCP Console, in the Groups tab. If you can’t see the Groups tab, follow that URL, and it’ll prompt you to select your organisation’s account (rather than your project). Then follow the prompts to add a new account to a group, paste your account’s email address, set appropriate permissions, and it’ll work!