Extend a $C^2$ function from the Boundary to the Interior while Preserving Some Regularity

Question

Let $U$ be some open connected bounded set in $\mathbb{R}^n$ with $C^2$ boundary and $g \in C^2(\partial U)$. Can we extend $g$ to the interior of $U$ such that ${extension}(g) \in W^{2, 2}(U)$? I am aware of the Whitney's Extension Theorem, but we have a much stronger hypothesis and weaker requirement here. Is there a way to do this while not using Whitney's Extension?