From Automorphism of code to automorphism of lattice

Question

From a code, a lattice can be constructed using many methods. For codes over $\mathbb{F}_2$, there is the straight construction \begin{equation} \Lambda(C) := \{v/\sqrt{2} \ | \ v \in \mathbb{Z}^n \ \land \ v = c + (2k_1, 2k_2,....,2k_n), c \in \mathcal{C}, k_{i} \in \mathbb{Z}\} \} \end{equation} and the twisted construction (for example from $G_{24}$ leech lattice is constructed using the twisted construction). Can someone explain if there is a general result regarding what happens to the Automorphism group, which are in general finite simple groups. For example, if I know the automorphism group of a code as Aut($\mathcal{C}$), what can one say about the automorphism group of Aut($\Lambda_1$) and Aut($\Lambda_2$), where $\Lambda_1$ and $\Lambda_2$ are obtained from $\mathcal{C}$ by straight and twisted construction respectively. Is it safe to say that Aut($\mathcal{C}$) $\subset$ Aut($\Lambda_i$), for $i$ = 1,2 ?