Question 1: Is the following statement true?
($*$) Let $\mathcal{L}$ be a first order language and $\mathcal{M}$ a $\mathcal{L}$-structure and $H\leq Aut(\mathcal{M})$ then there exists a substructure $\mathcal{N}\subseteq\mathcal{M}$ such that $H\cong Aut(\mathcal{N})$
Question 2: If the answer of question 1 is negative, what are examples of sufficient conditions on $\mathcal{L}$ and $\mathcal{M}$ such that ($*$) holds? Is there any intuitive necessary and sufficient condition?
The answer to Question 1 is negative. Suppose the language has no non-logical symbols (or just equality, if you consider equality non-logical) and let $\mathcal M$ be the structure consisting of a $3$-element set. Then its automorphism group is the symmetric group on those $3$ elements. Let $H$ be the alternating subgroup, i.e., the unique, cyclic subgroup of order $3$. This is not isomorphic to the automorphism group of any substructure of $\mathcal M$. The automorphism group of a substructure is either trivial or of order $2$, according as the substructure has $1$ or $2$ elements.
I don't have an answer to Question 2, except for the triviality that rigidity of $\mathcal M$ gives a vacuously sufficient condition.