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?