  Given a model $\mathcal{M}$ of set theory, and a nontrivial automorphism $j$
of $\mathcal{M}$, let $\mathcal{I}_{\mathrm{fix}}(j)$ be the submodel of
$\mathcal{M}$ whose universe consists of elements $m$ of $\mathcal{M}$ such
that $j(x)=x$ for every $x$ in the transitive closure of $m$ (where the
transitive closure of $m$ is computed within $\mathcal{M}$). Here we study the
class $\mathcal{C}$ of structures of the form $\mathcal{I}_{\mathrm{fix}}(j)$,
where the ambient model $\mathcal{M}$ satisfies a frugal yet robust fragment of
$\mathrm{ZFC}$ known as $\mathrm{MOST}$, and $j(m)=m$ whenever $m$ is a finite
ordinal in the sense of $\mathcal{M}$. We show that every structure in
$\mathcal{C}$ satisfies
$\mathrm{MOST}+\Delta_0^\mathcal{P}\textrm{-Collection}$. We also show that the
following countable structures are in $\mathcal{C}$: (a) transitive models of
$\mathrm{MOST}+\Delta_0^\mathcal{P}\textrm{-Collection}$, (b) recursively
saturated models of $\mathrm{MOST}+\Delta_0^\mathcal{P}\textrm{-Collection}$,
(c) models of $\mathrm{ZFC}$. It follows from (b) that the theory of
$\mathcal{C}$ is precisely $\mathrm{MOST+\Delta}_{0}^{\mathcal{P}}$-Collection.
We conclude by proving a refinement of a result due to Amir Togha.
