Let $S$ be a semigroup and let $T$ be a subsemigroup of $S$. Then $T$ acts on $S$ by left and by right multiplication. This gives rise to a partition of the complement $S \setminus T$ and to each equivalence class of this partition we naturally associate a relative Schützenberger group. We show how generating sets for $S$ may be used to obtain generating sets for $T$ and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for $S$ from given presentations of $T$ and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity, finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups). These results provide common generalisations of several classical results from group theory and Rees index results from semigroup theory.