This note is concerned with the following question: What is the structure of those groups which possess two antiautomorphisms different from identity such that every element of the group is fixed by (at least) one of them?

C. Ayoub [1] stated this problem after having proved a statement equivalent to the following: The group G is a non-abelian extension of an abelian group by a group of order two, if, and only if, there is an automorphism α ≠ 1 and an antiautomorphism β ≠ 1 such that every element of G is fixed by a α by β. The Theorem at the end of the paper will show that the class of groups considered by C. Ayoub coincides with the class considered here.