  We continue the study of the virtual large cardinal hierarchy by analysing
virtual versions of superstrong, Woodin, and Berkeley cardinals. Gitman and
Schindler showed that virtualizations of strong and supercompact cardinals
yield the same large cardinal notion. We provide various equivalent
characterizations of virtually Woodin cardinals, including showing that On is
virtually Woodin if and only if for every class A, there is a proper class of
virtually A-extendible cardinals. We introduce the virtual Vopenka principle
for finite languages and show that it is not equivalent to the virtual Vopenka
principle (although the two principles are equiconsistent), but is equivalent
to the assertion that On is virtually pre-Woodin, a weakening of virtually
Woodin, which is equivalent to having for every class A, a weakly virtually
A-extendible cardinal. We show that if there are no virtually Berkeley
cardinals, then On is virtually Woodin if and only if On is virtually
pre-Woodin (if and only if the virtual Vopenka principle for finite languages
holds). In particular, if the virtual Vopenka principle holds and On is not
Mahlo, then On is not virtually Woodin, and hence there is a virtually Berkeley
cardinal.
