Cyclic-quasi-injective for Finite Abelian Groups

We investigate the conditions for a finite abelian group $G$ under which any cyclic subgroup $H$ and any group homomorphism $f \in \operatorname{Hom}(H,G)$ can be extended to an endomorphism $F \in \operatorname{End}(G)$. As a result, we provide necessary and sufficient conditions for such a group $G$ and we compute the number of cyclic subgroups possessing non-extendable homomorphisms. In addition, we demonstrate that the number of cyclic subgroups that do not satisfy the conditions corresponds to the sum of the maximum jumps in the associated permutations given by $\sum_{\sigma \in S_{n}} \max_{1 \leq i \leq n} \{\sigma(i) - i\}$.