I have a custom field on a custom object, that field needs to be editable for a public group and for rest of the users/profile this should be read only.
How would I manage to achieve this.
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Groups are associated with data visibility (roles, sharing etc.) whereas you're talking about field level security which is associated with profiles, so this approach won't work easily. You would have to either rely on page layouts or complex control through Apex, neither of which I'd recommend.
What you should do is create a profile with the base level of access required on those fields i.e. read-only, and they use permission sets to grant the requisite users write access.