Access as in registered, special, etc? No, the purpose of the integration is to allow access to the forum category based on what groups they're in.
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Only group members have access to group forums, and I believe this is what you are asking.
At this time, the only way to access the group forum is from the group page, especially if the parent Forum is either unpublished or access to Super Admins only as suggested.
If I visit a public group that I am not a member of and click the Forum link, I receive a "You do not have access to this forum" message
This isn't the case for me at all. Anyone clicking on "Group forum" in the group page can see the forum. Given there is currently no access control support within Kunena I can't imagine how this is working for you, unless some recent template update is doing it prior to Kunena access?
Even then it should still be a reasonably trivial matter for a user to recode a URL to access a private group forum.
I've implemented a solution using MetaMod that redirects unauthorised members -