Let G and H be locally compact groups. We will show that each contractive Jordan isomorphism Φ: L1(G) → L1(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.The authors were supported by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe” Grant PID2021-122126NB-C31 and by Junta de Andalucía Grant FQM185
