In $V$, let me call a set theory structure A is a $\omega_1$ model if the $\omega_1$ of $A$ is the same as the $\omega_1$ in $V$ (up to isomorphism). The question I would like to ask is the following: Given any transitive structure $(B,\in)$ containing $\omega_1$ in $V$, is it always possible to find an $\omega_1$ model $A$ such that $B\prec A$ and $B\cap H(\omega_2)$ is in the well-founded part of $A$.

Some remark: Lowenheim-Skolem or Compactness theorem can be employed to construct an extension. But in general this extension is not an $\omega_1$ model. To get one $\omega_1$ model, one can either use some large cardinal or force a generic ultrafilter to construct an ultrapower which is well founded below $\omega_1$. But I am not sure how to get this using ZFC alone. On the other hand, it is known that the existence of $\omega$-model is correct between transitive ZFC models and thus can be constructed under ZFC.

I know very little about model theory of set theory. So it would be very helpful if anyone can provide some reference about this topic.