  We investigate the automorphisms of some $\kappa$- existentially closed
groups. In particular, we prove that $Aut(G)$ is the union of subgroups of
level preserving automorphisms and $|Aut(G)|=2^{\kappa}$ whenever $\kappa$ is
inaccessible and $G$ is the unique $\kappa$-existentially closed group of
cardinality $\kappa$. Indeed, the latter result is a byproduct of an argument
showing that, for any uncountable $\kappa$ and any group $G$ that is the limit
of regular representation of length $\kappa$ with countable base, we have
$|Aut(G)|=\beth_{\kappa+1}$, where $\beth$ is the beth function. Such groups
are also $\kappa$-existentially closed if $\kappa$ is regular. Both results are
obtained by an analysis and classification of level preserving automorphisms of
such groups.
