Dear All! I guess this should be well-known, but I cannot find it in literature:
Is it true that if $H$ is a subgroup of finite index in $G$, and $H$ is hopfian, then $G$ is also hopfian?
In case when the groups are finitely generated, the answer is "yes" by R. Hirshon.
(And the other way around question has answer no, as was kindly answered by Kate Juschenko, even for finitely generated case -- which is quite bizarre!!!)