I have recently stumbled across the problem of describing the characteristic subgroups of a finite abelian group. With some discussions with some mathematicians in my lab, I managed to obtain a "visual" description of such, and wrote it down.

Nonetheless, I know that some of this work has already been clear by Kaplansky in "Infinite abelian groups". Combining both theorems 25 and 26, we obtain a description for every finite abelian $p$-group for $p\neq 2$, and then for every abelian finite group of odd order. I managed to obtain a (less aesthetical, but still understoodable) description for the finite abelian $2$-groups that Kaplansky mentions in the theorem 27 as problematic. In this book, I didn't manage to find some reference for such groups, but maybe some person here knows an article that has already tackles this question.

So my question is : has some text already given a reasonable parametrization of characteristic subgroups of finite abelian $2$-groups ?