Let $V$ be a model of $ZFC+GCH$ and let $V[G]$ be a generic extension of $V$ in which $CH$ fails.
Question 1. Is there a model $W$ such that:
1) $V \subseteq W \subseteq V[G],$
2) $W\models CH,$
3) $W$ and $V[G]$ have the same cardinals.
Question 2. The same question as above, this time assuming $V=L$ and there are no large cardinals?