  We describe the groups of automorphisms of two generated free braided
associative algebras with involutive diagonal braidings over a field of
characteristic $\neq 2$. Depending on the form of the diagonal involutive
braiding, five different automorphism groups arise as automorphism groups of
two generated free braided associative algebras. This list covers all three
automorphism groups that arise in the case of quantum planes \cite{AC}.
