Let $\Gamma$ be a model-theoretic interpretation of a structure $B$ in a structure $A$. Then $\Gamma$ induces a group homomorphism $\alpha_\Gamma:\mathrm{Aut}(A) \rightarrow \mathrm{Aut}(B)$. (See, for example, the subsection The associated functor, Section 4.3, in Shorter model theory by Hodges).
Suppose $B = A$; then $\alpha_\Gamma$ is a map $\mathrm{Aut}(A) \rightarrow \mathrm{Aut}(A)$. I suspect that this is bijective. Is this true? If so, what are possible hints for a proof of the fact?
No, it's not true. The idea is that it's possible to interpret a structure $A$ in some smaller part of itself, so that the interpreted copy of $A$ doesn't have all the information about the interpreting copy of $A$.
For example, let $K = \{P_i\mid i\in\omega\}$, all unary predicate symbols, and let $A$ be a countable structure which is partitioned by the $P_i$ into subsets of some equal size $>1$. Let $A'$ be an isomorphic copy of $A$. We interpret $A'$ in $A$ by taking $\text{dom}(A')$ to be defined by $\lnot P_0(x)$ in $A$, and $P_i(x)$ in $A'$ to be defined by $P_{i+1}(x)$ in $A$.
Now the map $\text{Aut}(A) \to \text{Aut}(A')$ induced by this interpretation just takes an automorphism of $A$ and restricts it to the domain of $A'$. This is not injective: its kernel is the permutations of $P_0(A)$ which fix the rest of the structure.