Is there an omitted "that"? No. The phrasing "a group that unlikely to be . . ." makes no sense.
Does the phrase "unlikely to be . . ." modify "a group"? Yes. In spite of the -ly ending, "unlikely" is an adjective. On its own, it would be naturally placed before the noun: "an unlikely group". That natural placement isn't a required placement. We can find adjectives following the nouns that they modify in poetry and song, in old-fashioned text and certain fixed phrases, and in cases where some other element of syntax makes it convenient. In this case, the adjective is modified by the infinitive phrase "to be representative of the group at issue".* Since the infinitive phrase needs to follow the adjective that it modifies, the entire adjective phrase follows the noun that it modifies.
Is "a group" different from "the group". Yes. The phrase "the group" is part of the phrase "the group at issue". The phrase "a group" is part of the phrase "a group unlikely to be representative of the group at issue". These are two different groups. Whatever the issue happens to be, "the group" represents the entire group which is definitively involved with the issue, but "a group" represents some group that isn't involved, or at least isn't involved in the same way. That the two groups are different is the main point of the phrasing. The author's opinion is that "a group" is too different from "the group" to be useful.
* It is most likely that "in the conclusion" modifies "relies". It is possible that it modifies "issue", "the group at issue", or even "representative", any of which would make it part of the infinitive phrase. That determination cannot be made without further context.