I am interested in the following property, be it on an abstract or concrete category:

$A$ is a substructure of $B$ such that every automorphism of $A$ extends **uniquely** to an automorphism of $B$. Or we can speak of an embedding $\iota:A\rightarrow B$ such that for every automorphism $\alpha$ of $A$ there exists a **unique** automorphism $\beta$ of $B$ such that $\iota\alpha=\beta\iota$.

This condition is not too difficult to show up in many combinatorial, algebraic and geometrical settings, at least. So, I wonder if it has been named already.