  An inverse semigroup $S$ is a semigroup in which every element has a unique
inverse in the sense of semigroup theory, that is, if $a \in S$ then there
exists a unique $b\in S$ such that $a = aba$ and $b = bab$. We say that an
inverse semigroup $S$ is a homogeneous (inverse) semigroup if any isomorphism
between finitely generated (inverse) subsemigroups of $S$ extends to an
automorphism of $S$. In this paper, we consider both these concepts of
homogeneity for inverse semigroups, and show when they are equivalent. We also
obtain certain classifications of homogeneous inverse semigroups, in particular
periodic commutative inverse semigroups. Our results extend both the
classification of homogeneous semilattices and the classification of certain
classes of homogeneous groups, in particular the homogeneous abelian groups and
homogeneous finite groups.
