  Testing isomorphism of infinite groups is a classical topic, but from the
complexity theory viewpoint, few results are known. S{\'e}nizergues and the
fifth author (ICALP2018) proved that the isomorphism problem for virtually free
groups is decidable in $\mathsf{PSPACE}$ when the input is given in terms of
so-called virtually free presentations. Here we consider the isomorphism
problem for the class of \emph{plain groups}, that is, groups that are
isomorphic to a free product of finitely many finite groups and finitely many
copies of the infinite cyclic group. Every plain group is naturally and
efficiently presented via an inverse-closed finite convergent length-reducing
rewriting system. We prove that the isomorphism problem for plain groups given
in this form lies in the polynomial time hierarchy, more precisely, in
$\Sigma_3^{\mathsf{P}}$. This result is achieved by combining new geometric and
algebraic characterisations of groups presented by inverse-closed finite
convergent length-reducing rewriting systems developed in recent work of the
second and third authors (2021) with classical finite group isomorphism results
of Babai and Szemer\'edi (1984).
