  Let $G$ and $H$ be locally compact groups. We will show that each contractive
Jordan isomorphism $\Phi\colon L^1(G)\to L^1(H)$ is either an isometric
isomorphism or an isometric anti-isomorphism. We will apply this result to
study isometric two-sided zero product preservers on group algebras and,
further, to study local and approximately local isometric automorphisms of
group algebras.
