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]: http://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