The automorphisms of Petit's algebras

Let sigma be an automorphism of a field K with fixed field F. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t;sigma]/fK[t;sigma] obtained when the twisted polynomial fK[t;sigma] is invariant, and were first defined by Petit. We compute all their automorphisms if sigma commutes with all automorphisms in Aut(F)(K) and nm-1, where n is the order of sigma and m the degree of f, and obtain partial results for n<m-1. In the case where K/F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F. We also briefly investigate when two such algebras are isomorphic.