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 $B$.

Three nice answers:
(1) [This question][1] deals with $C^*$-algebras. 
(2) [This paper][2] deals with extending involutions on Frobenius algebras.
(3) [A counter-example][3] for Boolean 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.


  [1]: https://mathoverflow.net/questions/192281/inner-and-extendible-automorphisms-of-c-algebras
  [2]: https://link.springer.com/article/10.1007/s002290200276
  [3]: https://math.stackexchange.com/questions/1406080/extending-automorphisms-in-complete-boolean-algebras