Hello, I have the following question: Is it possible for a complete $L_{\omega_1,\omega}$ sentence $\phi$ to satisfy (a) the (unique) countable model of $\phi$ has $2^{\aleph_0}$ many automorphisms and (b) there is a model $N$ of $\phi$, $N$ of size $\aleph_\omega$ and $N$ has $\le\aleph_\omega$ many automorhphisms?

So, $\phi$ will have "many" automorphisms in the countable case and "not very many" automorphisms for a model in power $\aleph_\omega$. Is this possible and if so, can you reference any examples?