(I ran into this question while thinking about the (Strong) Inner Model Hypothesis: see http://www.jstor.org/stable/4093051 or http://arxiv.org/abs/0711.0680.)

A *measurable cardinal* is a cardinal $\kappa$ such that there is an elementary embedding $j: V\rightarrow M\subseteq V$ with $M$ an inner model of $V$ and $crit(j):=\min\lbrace \alpha\in ON: j(\alpha)\not=\alpha\rbrace=\kappa$.

Now we can allow the target model of $j$ to live, not inside $V$, but inside some set-generic extension of $V$ as follows. Say that $\kappa$ is *outer-measurable* if there is some poset $\mathbb{P}\in V$, some $G$ which is $\mathbb{P}$-generic over $V$, and some transitive inner model $M$ of $V[G]$ such that there is an elementary embedding $$ j: V\rightarrow M\subseteq V[G]$$ with $crit(j)=\kappa$.

In general, given any large cardinal property $(\*)$ defined in terms of elementary embeddings, we can define *outer-$(\*)$*-ness to be the property $(\*)$ where the target model $M$ is allowed to be an inner model of some set-forcing extension of $V$, rather than $V$ itself. My questions, then, are:

Is there a large cardinal property $(\*)$ such that we can have an outer-$(\*)$ cardinal which is not $(\*)$?

Is there a large cardinal property $(\*)$ such that the consistency strength of an outer-$(\*)$ cardinal is weaker than the consistency strength of a $(\*)$-cardinal?

I suspect that the answer to the second question is "no;" I have no idea about the first question.

[EDIT: Thanks to Joel for pointing out that my now-removed claim that "measurable=outer-measurable" is wrong.]