If you'd like to post something in a Facebook group without revealing your name, use Facebook's anonymous posting feature. It's easy to do, assuming that a group administrator has enabled the feature.
Meta is introducing a new identity option in Facebook Groups: a 'nickname' feature that lets users participate under a custom ...
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Facebook adds nicknames in Groups to take on Reddit

Facebook is rolling out a new nicknames feature in Groups that lets people post under an alias instead of their full profile name, a clear attempt to capture some of the pseudonymous culture that has ...
Facebook Groups now let users post with usernames, offering more privacy while keeping a consistent identity across posts and ...