I haven't seen a big treatment. My source for Ackermann's set theory is Azriel Levy's article referenced at the very end of the review of the book on Finsler set theory which is found on my Web page. That article has further references to articles in which the equiconsistency with ZF is shown. I can't remember what I wrote to you about Ackermann's theory. The objects of our world are classes, on which there is a membership relation.
Some classes are sets.
Classes with the same elements are equal. Subclasses of sets are sets. Any condition on sets defines a class. Any condition on classes which does not mention the sethood predicate or have any non-set parameters, and which turns out to be true only of sets, defines a set. There is an infinite set.