  This paper investigates the preservation of hopficity and co-hopficity on
passing to finite-index subsemigroups and extensions. It was already known that
hopficity is not preserved on passing to finite Rees index subsemigroups, even
in the finitely generated case. We give a stronger example to show that it is
not preserved even in the finitely presented case. It was also known that
hopficity is not preserved in general on passing to finite Rees index
extensions, but that it is preserved in the finitely generated case. We show
that, in contrast, hopficity is not preserved on passing to finite Green index
extensions, even within the class of finitely presented semigroups. Turning to
co-hopficity, we prove that within the class of finitely generated semigroups,
co-hopficity is preserved on passing to finite Rees index extensions, but is
not preserved on passing to finite Rees index subsemigroups, even in the
finitely presented case. Finally, by linking co-hopficity for graphs to
co-hopficity for semigroups, we show that without the hypothesis of finite
generation, co-hopficity is not preserved on passing to finite Rees index
extensions.
