TITLE: Elementary Embeddings and Relative Constructibility
QUESTION [7 upvotes]: Suppose $$j:M\prec N$$ is a non-trivial elementary embedding. Under what conditions on the sets (classes?) $M$ and $N$ (or even the critical point of $j$) does $j$ extend to an elementary embedding $$k:L(M)\prec L(N)?$$ 
In the case where $M=N=V_{\lambda+1}$, the existence of such a $k$ is a strictly stronger assumption (this is Woodin's $I_0$ axiom), but need this always be the case? 
A more specific question involves extendible cardinals: Recall that $\kappa$ is extendible if, for every $\eta >\kappa$ there exists a $\theta>\eta$ and an elementary embedding $j:V_{\eta+1}\prec V_{\theta+1}$ such that $crit(j)=\kappa$ and $j(\kappa)>\eta$. Does $j$ extend to a $$k:L(V_{\eta+1})\prec L(V_{\theta+1})?$$
Is the existence of such a $k$ a straight-forward construction or is it strictly stronger than the existence of an extendible cardinal?

REPLY [4 votes]: Not all such embeddings extend, as was mentioned at Bob Lubarsky's question on Extending elementary embeddings from initial segments to all of $V$. 
Specifically, suppose that $\kappa$ is the least $1$-extendible cardinal. So there is an elementary embedding $j:V_{\kappa+1}\to V_{\eta+1}$. I claim that this embedding cannot extend to $L(V_{\kappa+1})\to L(V_{\eta+1})$, since the $j\upharpoonright V_{\kappa+1}$ is determined by $j''V_{\kappa+1}$, which is a size $2^\kappa$ subset of $V_{\eta+1}$, and by means of a flat pairing function all such subsets are coded as elements of $V_{\eta+1}$. Thus, $L(V_{\eta+1})$ would see that $\kappa$ is $1$-extendible, and so by elementarity there must be a $1$-extendible cardinal in $L(V_{\kappa+1})$ below $\kappa$, contradicting the minimality of $\kappa$. 
A similar argument applies to the least $\theta+1$-extendible cardinal for any $\theta$.