Full disclosure: I am an IT professional responsible for the proper care and feeding of servers and systems that lots of users rely on.
While some of this is tangential, I think it bears expanding here:
In a best-case scenario, if Share A should only be accessible to User Group A and NOT to User Group B, then the permissions on the share AND the files in it should be set to both Include User Group A and Exclude User Group B. The exclusion is usually best done by excluding everyone, and only including the specific User Group A
In many cases, what is done instead is that the Share A is open for people looking for it, but is only mapped for people in User Group A and the Administrator, either intentionally or unintentionally is simply betting that the other users won't go looking for what else is out there.
This is not correct, it is asking for trouble, but it is also the reality in far too many places.
OP MAY have been one of these User Group B people, disallowed by company policy but not by actual technology permissions and settings, from accessing Share A.
Another issue is that access is determined or designed to be in a particular way. While is it in some way commendable that OP is trying to get LESS access than possible, there may be reasons why this is inadvisable and if they actually NEED less access than they have been granted, this should also be a request to their IT staff.
In a properly configured environment people in User Group B should either not even see Share A or, if they do, they should be denied access to it.
There is a proper balance between giving good answers to good questions, but as a long-time user of this site, both for my own questions and for answers, and as an IT professional, I agree with the generally strict application of the No Corporate IT rule. There are simply too many variables, too many issues we cannot know the full context of, and frankly, most local IT should be able to answer most such questions far quicker and with far more authority than we can here. Further, to answer questions here is to subvert local IT. There are plenty of local IT people I've come across who are difficult to work with and that is sad, but at least as many times I've seen that the fault in tense user-IT relations cannot be laid solely on IT, and unwilling and uncooperative users may be at least as much to blame for difficulties in that relationship. Giving these users a backdoor support desk is NOT something that is actually helpful over the long term. While it may be only implied, a business hiring local IT assumes they expect their users to rely on them for their support needs. Enabling the circumvention of this is akin to assisting with circumvention of license requirements (not allowed here) and legal requirements (also not allowed).