If I have an (abelian) group $G$ and an automorphism $\sigma: G \to G$ then for any subgroup $H$ of $G$ there is an induced isomorphism $G/H \cong G/\sigma(H)$ given by the map $gH \mapsto \sigma(g)\sigma(H)$.

I want to know if the converse statement is true. Suppose I have an (abelian) group $G$ and two subgroups $H_1$ and $H_2$ such that $G/H_1 \cong G/H_2$ via some isomorphism. Then is there an automorphism $\sigma: G \to G$ which induces the automorphism on the quotient? I'm also interested in the slightly weaker question of if there is always an automorphism so that $\sigma(H_1) = H_2$ (ie an automorphism which induces some potentially different isomorphism on the quotients).

This feels plausible to me (I can't find a counterexample), but I haven't been able to construct the automorphism in general.