Let $S$ be a semigroup and let $T$ be a subsemigroup of $S$. Then $T$ acts on $S$ by left and by right multiplication. If the complement $S \setminus T$ has finitely many strong orbits by both these actions we say that $T$ has finite Green index in $S$. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents.