Let $A \subseteq B$ be two (associative with $1$) $k$-algebras, where $k$ is a field of characteristic zero, and let $f$ be a $k$-automorphism of $A$.

I am interested to know 'when' one can extend $f$ to a $k$-automorphism of $B$.

Two nice answers: (1) This question deals with $C^*$-algebras. (2) This paper deals with extending involutions on Frobenius algebras.

Since my question is too general, I do not mind to concentrate on commutative algebras only (which are not fields), and $f$ of finite order (for example, $f$ is an involution, namely, of order $2$).

Thank you very much for any comment.