In fact a **substantially stronger** embedding is true:

> **Theorem.** If $n\geq 2$, $p>1$, and $\Omega\subset\mathbb{R}^n$ is a bounded domain with Lipschitz boundary, then
> $W^{1,p}(\partial\Omega)\subset W^{1-\frac{1}{q},q}(\partial\Omega)$,
> where $q=\frac{np}{n-1}$. That is, there is a bounded linear extension
> operator $$ E:W^{1,p}(\partial\Omega)\to W^{1,q}(\Omega)\cap
 C^\infty(\Omega). $$

When $p=2$ i.e., $W^{1,2}(\partial\Omega)=H^1(\partial\Omega)$, we have $q=\frac{2n}{n-1}>2$. Hence the extension is
$$
E:H^1(\partial\Omega)\to W^{1,q}(\Omega)\subset H^1(\Omega),
$$
proving (indirectly) that $H^1(\partial\Omega)\subset H^{\frac{1}{2}}(\partial\Omega)$ (because traces of $H^1(\Omega)$ are in $H^{\frac{1}{2}}(\partial\Omega)$.

The above theorem is known, but it is hard to find a reference to an elementary proof of it. I will add references later.