  In this paper, we investigate *-homomorphisms between C*-algebras associated
to \'etale groupoids. First, we prove that such a *-homomorphism can be
described by closed invariant subsets, groupoid homomorphisms and cocycles
under some assumptions. Then we prove C*-rigidity results for \'etale groupoids
which are not necessarily effective. As another application, we investigate
certain subgroups of the automorphism groups of groupoid C*-algebras. More
precisely, we show that the groups of automorphisms that globally preserve the
function algebras on the unit spaces are isomorphic to certain semidirect
product groups. As a corollary, we show that, if group actions on groupoid
C*-algebras fix the function algebras on the unit spaces, then the actions
factors through the abelianizations of the acting groups.
