Let $M$ be a countable structure that is homogeneous, i.e. every isomorphism between finitely generated substructures of $M$ extends to an automorphism of $M$. Let $\phi_M$ be the Scott sentence of $M$.
If $N$ is an uncountable model of $\phi_M$, then $N$ is not necessarily homogeneous, but is the existence of an uncountable model sufficient for the existence of an uncountable homogeneous model? If so, can we further conclude that given $N\models\phi_M$, $N$ uncountable, there is a homogenous model of $\phi_M$ of the same cardinality as $N$?