  In [21] it was asked if equality on the reals is sharp as a lower bound for
the complexity of topological isomorphism between oligomorphic groups. We prove
that under the assumption of weak elimination of imaginaries this is indeed the
case. Our methods are model theoretic and they also have applications on the
classical problem of reconstruction of isomorphisms of permutation groups from
(topological) isomorphisms of automorphisms groups. As a concrete application,
we give an explicit description of $\mathrm{Aut}(\mathrm{GL}(V))$ for any
vector space $V$ of dimension $\aleph_0$ over a finite field, in affinity with
the classical description for finite dimensional spaces due to Schreier and van
der Waerden.
