Suppose $\kappa$ carries an $\omega_1$-saturated countably complete$\kappa$-complete ideal $I$, given $M\prec (V_{\kappa+2},\in , <)$ ($<$ well orders $V_{\kappa+2}$) of size $<\kappa$ containing $I$, we show how to find an end-extension $N\prec V_{\kappa+2}$ of $M$ (i.e. $N\cap \sup(M\cap \kappa)=M\cap \kappa$).
Let $G\subset P(\kappa)/I$, and let $j: V\to N$ be the generic embedding in $V[G]$. Look at $M'=\{j(f)(\kappa): f\in M\}$. In $N$, $M'\prec j((V_{\kappa+2}, \in , <))$. Clearly $\kappa\in M'\cap j(\kappa)- j(M)$. Note $j(M)\cap j(\kappa)=j''M \cap j(\kappa)=j(M\cap \kappa)=j'' M\cap \kappa=M\cap \kappa$. If $\xi\in \kappa$ such that $\xi\in M'$, $\xi\in M$. The reason is that: fix $f$ such that $j(f)(\kappa)=\xi$. In $V$, $M$ contains a maximal antichain $\{A_i: i<\omega\}$ such that for each $i$, there exists $\delta_i<\kappa$, $A_i\Vdash j(f)(\kappa)=\delta_i$. Clearly $\{A_i: i<\omega\}, \{\delta_i: i<\omega\}\subset M$. Hence in $V[G]$, there exists $i<\omega$, $\xi=\delta_i \in M$. This shows $M'\cap \sup (j(M)\cap j(\kappa))=M'\cap \sup(M\cap \kappa)=M\cap \kappa=j(M)\cap j(\kappa)$.
By elementarity, there exists $N\prec V_{\kappa+2}$ containing $M$ such that $N\cap \sup (M\cap \kappa)=M\cap \kappa$ and $N\cap \kappa \neq M\cap \kappa$.
So $\kappa$ can be $2^\omega$.