Flag-transitive $2$-designs with block size 5 and alternating groups

This paper contributes to the classification of flag-transitive 2-designs with block size 5. In a recent paper, the flag-transitive automorphism groups of such designs are reduced to point-primitive groups of affine type and almost simple type, and a classification is given of such automorphism groups with sporadic socle. In the present paper, we classify such designs admitting a flag-transitive automorphism group whose socle is an alternating group. We prove that there are precisely six such designs and determine the corresponding automorphism groups.