My apologies: There were a couple of typos in the original question. Hope I got them all.

Let $\kappa$ be an uncountable cardinal of cofinality $\omega$ and $M$ a model of size $\kappa$. We equip $\kappa^\kappa$ with the product topology and let $Aut(M)\subset \kappa^\kappa$ denote the topological group of automorphisms of $M$. The closure of $Aut(M)$ under the product topology is denoted by $\overline{Aut(M)}$ and $\partial Aut(M)=\overline{Aut(M)}\setminus Aut(M)$ denotes the boundary of $Aut(M)$.

My question is: If we know that (EDIT)$|\partial Aut(M)|\ge\kappa^+$(EDIT), can we say anything about the cardinality of $Aut(M)$, other than (EDIT)$|Aut(M)|\le|\partial Aut(M)|$(EDIT)?

Side note: If $f$ is in $\partial Aut(M)$, then $f$ is 1-1, not onto, and for every formula $\phi$ and every finite $\vec{a}$, $M\models\phi[\vec{a}]$ iff $M\models\phi[f(\vec{a})]$. I.e. $f$ is an elementary embedding.