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 $\operatorname{Aut}(M)\subset \kappa^\kappa$ denote the topological group of automorphisms of $M$. The closure of $\operatorname{Aut}(M)$ under the product topology is denoted by $\overline{\operatorname{Aut}(M)}$ and $\partial \operatorname{Aut}(M)=\overline{\operatorname{Aut}(M)}\setminus\operatorname{Aut}(M)$ denotes the boundary of $\operatorname{Aut}(M)$.
My question is: If we know that (EDIT)$|\partial\operatorname{Aut}(M)|\ge\kappa^+$(EDIT), can we say anything about the cardinality of $\operatorname{Aut}(M)$, other than (EDIT)$|\operatorname{Aut}(M)|\le|\partial\operatorname{Aut}(M)|$(EDIT)?
Side note: If $f$ is in $\partial\operatorname{Aut}(M)$, 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.