Saturation implies homogeneity

A $\kappa$-saturated model has the one-point extension property for partial elementary maps with domain of size less than $\kappa$. Given $f:A\to B$ and $a$, transport the complete type of $a$ over $A$ along $f$. Every finite subset is satisfiable by elementarity of $f$, so saturation realizes the transported type by some $b$. Then $f\cup \{(a,b)\}$ is a partial elementary map; inequalities in the type ensure that a new $a$ gets a new image.

When the model itself has infinite cardinality $\kappa$, the back-and-forth method upgrades this local property to automorphisms extending all partial elementary maps of size less than $\kappa$. Enumerate the model in $\kappa$ stages and alternately extend the domain and range. Every intermediate map still has size less than $\kappa$. For a model of larger cardinality, $\kappa$-saturation alone gives the local extension property; full automorphism extension requires additional size or homogeneity hypotheses.