= 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.
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