As clear from the title itself, I want to know if there are any characterizing properties (of say, orders and their prime divisors, on elementary divisors, invariant factors, composition series etc.) which classify the finite abelian groups $G$ for which there exist $n \in \mathbb Z_{> 2}$ and an (abelian) group $H$ such that $G \cong (\mathbb Z / n \mathbb Z)^\times \oplus H$. It is clear that this is equivalent to having $G \cong (\mathbb Z / p^k \mathbb Z)^\times \oplus H \cong \mathbb Z / p^{k-1} \mathbb Z \oplus \mathbb Z / (p-1)\mathbb Z\oplus H$ for some prime $p$ and $k \in \mathbb N$ (one direction being tautological and the other being a consequence of the Chinese Remainder Theorem). Any thoughts/ideas, suggestions, (partial or complete) results in that direction, proofs or references would be really appreciated.
Edit: I also just discovered this (Which finite groups are the group of units of some ring?) post, although that asks which groups are exactly equal to the group fo units of some ring. The condition I want is much weaker (at least as far as I can see) and I think it should probably have a much more elementary answer.
