Ebeling(Proposition 3.6)
In order to formulate the next result,
Step 1. We observe that an isomorphism $u : Λ_1 → Λ_2$ of lattices induces an isomorphism $u: Λ^*_1 → Λ^*_2$ of the dual lattices and determines an isomorphism $u' : q(Λ_1)→ q(Λ_2)$ of their discriminant quadratic forms.
Step 2. There is an induced homomorphism $O (Λ) → O ( q_Λ )$ between the automorphism groups of $Λ$ and $q_Λ$ .
Proposition 3.6. Two even overlattices $Λ → Γ$ and $Λ → Γ'$ are isomorphic if and only if the isotropic subgroups $H_Γ ⊂ G_Λ$ and $H_Γ' ⊂ G_Λ$ are conjugate under some automorphism of Λ .
Author left this theorem to the reader. But, i could not find any directions to approach this problem.