From Oxford Learner's Dictionaries...
countable + singular or plural verb
a number of people or things that are together in the same place or that are connected in some way
Example usage: A group of us is/are going to the theatre this evening.
Although it's not specifically mentioned there, I should point out that AmE has always tended to treat similar "composite entity" nouns (family, company, [the] government], [the] police,...) as singular in all contexts.
BrE is more flexible on this point, and thus probably more likely than AmE to treat group as plural in contexts where the relevant verb more logically attaches to each individual person or thing within the group, rather than to the group as a single entity. That's to say, in BrE we often treat a group as syntactically equivalent to members of a group if that's a better semantic match for the context.
In OP's specific context, the relativizer who only really works with a plural verb form (a group who damage reputations). With a singular verb, it would have to be a group that damages reputations.