Let $M$ be a countable transitive model of (enough of) ZFC. Mostowski's Absoluteness Theorem says that $\Pi^1_1$ statements are absolute between $M$ and larger models, in particular, between $M$ and the universe $V$.
For general $M$, this cannot be extended to $\Sigma^1_2$ statements, see Andrés Caicedo's answer here: Failure of Shoenfield's Absoluteness.
My question is: Can we have absoluteness of $\Sigma^1_2$ (and beyond) statements between $M$ and $V$ for some countable transitive models $M$? (Possibly at the expense of some local or global large cardinal assumptions.)
Or, is it the case that for any countable transitive model $M$, there is a $\Sigma^1_2$ (lightface) statement true in $V$ which fails in $M$?