Suppose $A$ and $B$ are subsets of $\mathbb{R}^2$ homeomorphic (and thus $C^\infty$ diffeomorphic) to the open annulus (punctured $\mathbb{B}^2$) and let $G$ be the group of isometries of ${\mathbb R}^2$ that preserve both $A$ and $B$ $\big(\phi (A)=A$ and $\phi (B)=B$, for any $\phi\in G \big)$.
Question: is there always a smooth diffeomorphism $f:A\rightarrow B$ such that $$f\circ \phi=\phi \circ f$$ for any $\phi\in G$, i.e, a $G$-equivariant $C^\infty$ diffeomorphism between $A$ and $B$ ?
The same question for $A$ and $B$ homeomorphic to $\mathbb{B}^2$ with finitely many points removed.