Given a model $\mathcal{M}$ with a domain $M$ and $B \subseteq M$, $B$ is an automorphism base for $\mathcal{M}$ iff $\forall f \in Aut(\mathcal{M}). (\forall b \in B. f(b)=b) \implies f = 1_\mathcal{M}$, that is every automorphism of $\mathcal{M}$ is uniquely determined by its restriction to $B$.
I have seen the interest (particularly in computability on degree structures) in studying the automorphism bases of some model $\mathcal{M}$ when one would like to find out the automorphism group $Aut(\mathcal{M})$. In particular I can understand that knowing small automorphism base for a given model, immediately bounds the size of the automorphism group, i.e. $\# Aut(\mathcal{M}) \le \# \mathcal{M}^{B}$. As one can focus only on the elements of $B$ instead of the whole $M$, many questions, e.g. a rigidity of $\mathcal{M}$ get simplified.
However, I would like to know the relation between the automorphism group of $\mathcal{M}$ and its reduct $\mathcal{B}$ with the domain $B$. I found out that it is not generally true that $Aut(\mathcal{B}) \hookrightarrow Aut(\mathcal{M})$ or $Aut(\mathcal{M}) \hookrightarrow Aut(\mathcal{B})$. However, one could perhaps say something on the generators and the defining relations of the two groups.
I wish I could make my question more precise, but to me right now it seems that there should be more reasons for studying the automorphism bases than the ones I stated. Could we at least say something more if we restrict the model $\mathcal{M}$ to be a partial order $(M,\le)$?