Construct vector field in $H^1$ with given divergence (and not only)

Question

I'm trying of understand if the following is true. Let $\Omega$ and $\Omega_0$ be some regular open bounded sets of $\mathbb R^d$ such that $\overline\Omega\subset\Omega_0$. I do not have constraints on the hypotheses on their boundaries. Let $q_0\in L^2(\Omega_0)$. Is it possible to construct $\mathbf v_0$ such that

1. $\operatorname{div}\mathbf v_0 = q_0$ in $\Omega_0$;
2. $\left.\mathbf v_0\right|_{\partial\Omega}\equiv 0$;
3. $\|\mathbf v_0 \|_{H^1(\Omega_0)} \le C \|q_0\|_{L^2(\Omega_0)}$.

The construction of a vector field satisfying 1. and 3. can be done by considering an associated Poisson problem with $q_0$ as right hand side and defining $\mathbf v_0$ as the gradient of the solution. Thus, 1. follows by construction and elliptic regularity implies 3. I'm wondering if it's possible to play around with Sobolev extensions from $\Omega$ to $\Omega_0$ so that 2. is satisfied.