Homeomorphism of the plane that takes a simple closed curve to a circle and is the identity off of a compact set

Question

Let $C$ be a simple closed curve in $\mathbb{R}^2$. Schoenflies Theorem gives a homeomorphism $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(C)$ is a circle. Suppose that $B$ is a closed ball containing $C$ in its interior. Is there a homeomorphism $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(C)$ is a circle and $f$ is the identity off of $B$?