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You could certainly use the REST API to make the actual changes, to your second site...but there is no automagic detection of "change" to your group on FirstSite. So you'd have to manage the change tracking yourself.
Is there any reason why you are using sites the way you are? Typically a user can get into FirstSite, but not SecondSite - the fact that your users/groups are exactly the same makes me wonder if you should combine these sites and use Projects to do your security...