"Him and his group" is grammatical here for some dialects of American English (and maybe others too).
One common explanation for this phenomenon is the linguistic idea of "default case", but first we have to explain the idea of "case".
Personal pronouns in English change form depending on where they are in the sentence; the first-person personal pronoun, for example, is "I" when it appears in (or "is licensed by") subject position, and is "me" everywhere else. This difference based on syntactic (word) position is one example of a widespread crosslinguistic phenomenon we call "case". We usually say that "I" is "Nominative" case, and "me" is either "accusative" or "oblique" case.
Anyway, "default case" is the idea that each language picks a case to appear in positions that aren't licensed by specific syntactic positions.
In Modern English (but not older English!) the default case appears to be the accusative/oblique. This is why, in English, the answer to the question "Who wants ice cream?" is "Me!" and not "I!"
In Spanish, however, where the default case is Nominative, the answer is "Yo", not "mí".
The fact that default case has switched from Nominative to Accusative in English has had a number of effects over the years. Answering the phone, for example, has gone from the archaic "This is she" to the more modern "this is her".
Copular/"be"-verb sentences like these are weird cross-linguistically anyway, and in this example the pronoun does not appear in a canonical subject position, which makes it much more likely to be affected/governed by default case. So, in older English we saw Nominative "this is she", but in Present Day English we often get Accusative/Oblique "this is her" in many dialects.
Another place where case is famously wonky is with conjunctions, especially "and". It seems that when a noun stands by itself, it always gets the expected case for that position (which, of course, is exactly what we would expect to happen), but when multiple nouns are joined by a conjunction in the same position, it's common for default case to be assigned instead, which has led to the idea that "case-assigning positions" (in this case subject position) "can't see down into the conjunction".
This leads to interesting case alternations like "I went to the store", but with a conjunction, "Me and him went to the store."
And this is exactly what we see here. In these dialects we get the expected Nominative case when the noun appears by itself, but we instead get default case when the conjunction blocks case assignment (or feature agreement, or however your preferred theory of syntax deals with case).
Interestingly, as a result of the pedantry seen in your comment and, more importantly, displayed by English teachers for at least the past hundred years, we've seen a series of interesting, unexpected results.
First was simple hypercorrection. After being corrected so long for saying "me and him" in subject position, the chastised speakers began changing "me and him" to "he and I" everywhere, not just in subject position.
This has led to sentences like "Jane went to the store with he and I" (where "him and me" is usually prescriptively considered "correct") becoming incredibly common in many dialects of American English.
Even further, and more interesting to me, however, is that this seems to have started becoming a marker of politeness, or even professional jargon.
For example, when writing work emails, I have to consciously make myself write a sentence like "Jane went to the store with him and me", because not using "he and I" feels unprofessional somehow, even though it's prescriptively "correct".
A similar example of corporate pronoun jargon weirdness under a conjunction can be seen in the use of "myself" instead of "me". For example, in a formal business meeting a person giving a presentation might say something like "If you have any questions, you can direct them either to Jane or myself", when according to prescriptive grammar rules "me" would be "correct" here.
Anyway, language is fun and weird, and its only constant is that it is always changing!