  Given countable abelian groups $C,A$, with $C$ torsion, we compute the
potential Borel complexity class of the classification problem
$mathcal{R}_{\mathbf{Ext}(C,A)}$ for extensions of $C$ by $A$. In particular,
we show that such a problem can have arbitrarily high potential complexity.
Furthermore, if $\Gamma $ is one of the possible potential Borel complexity
classes of the isomorphism relation for a Borel class of countable first-order
structures, then there exist countable torsion groups $C,A$ such that
$\mathcal{R}_{\mathbf{Ext}(C,A)}$ has potential Borel complexity class $\Gamma
$.
