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? 



Elementary Embeddings and Relative Constructibility
Not all such embeddings extend, as was mentioned at Bob Lubarsky's question on [Extending elementary embeddings from initial segments to all of $V$](https://mathoverflow.net/questions/146567/extending-elementary-embeddings-from-initial-segments-of-v-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 $\theta$. 


