Given any set $X$ and any $A,B\subseteq X$, is true that for every bijection $f:A\to B$ there exists $\phi\in\text{Sym}(X)$ such that $f={\phi|}_A$?

Question

Given any (not necessarily finite) set $X$ as well as arbitrary $A,B\subseteq X$, can we always find for every bijection $f:A\to B$ a permutation $\phi\in\text{Sym}(X)$ such that $f={\phi|}_A$ (the restriction of $\phi$ to $A$)?

I know that this is true when $X$ is finite, however what if $X$ is non-finite. Does it still hold then?

Answer 1

No: take $X=A=\mathbb{N}$, $B=\mathbb{N}\setminus \{ 27 \} $.