I think there are two main ways to see such objects: either as a group together with a chosen endomorphism (which is something you said in an earlier revision, and I think you might have kept that part because it greatly clarifies the notion), which is how most people would see them; or as group objects in the topos of $\mathbb{N}$-sets (which explains why they behave like groups in many ways). That being said, I don't think you can hope for much in way of classification.
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