Restricting extenders to a ground model Let V=W[g], where g is P-generic over W for some poset P in W. Let F be a V-extender with critical point κ such that P &in; VκW. If the support of F is sufficiently closed, say strength(F)=length(F)=λ for some inaccessible cardinal λ>κ, then F ∩W &in; W. To the best of my knowledge, this is due do Hamkins-Woodin; the argument for it which I have in mind is the one written up in  https://ivv5hpp.uni-muenster.de/u/rds/VM2_4.pdf . 
Can this be proven in more generality? I.e., what if we don't assume ult(V;F) to be Card(P)-closed, is it still true that F ∩W &in; W? Or is there a counterexample?
And can one even drop the hypothesis that F be a total V-extender, i.e., that it acts on some transitive model contained in V rather than on all of V? 
 A: Here is one way to make a kind of counterexample, although I'm not exactly sure if this is what you want.
Start in a model $W$ where $\kappa$ has two different normal measures $\mu$ and $\nu$. 
Now, consider the extension where we have added a Cohen real $V=W[c]$. Since this is small forcing, each measure $\mu$ and $\nu$ generates a measure in the extension $V$. Now, use the real $c$ to define a certain $\omega$-iteration of the (extensions of the) normal measures, where the digits of the real tell you which measure to use at each stage. Let $j:V\to M$ be the resulting iteration. This can be realized as an extender embedding with countably many generators. 
But the restriction $j\upharpoonright W$ cannot be an embedding in $W$ because from this restriction we can tell which measure we used at each stage and therefore we would be able to define $c$ in $W$, which contradicts that $c$ is $W$-generic.
Similar examples can be built for larger kinds of extenders. For example, if you have two different rank-to-rank embeddings $j_0:W_\lambda\to W_\lambda$ and $j_1:W_\lambda\to W_\lambda$, then in the extension $V=W[c]$, you can lift the original embeddings and use $c$ to define a certain iteration of them, and then use this to form an extender in $V$, which cannot lift any ground model embedding. 
