  Given a subset of states $S$ of a deterministic finite automaton and a word
$w$, the preimage is the subset of all states mapped to a state in $S$ by the
action of $w$. We study three natural problems concerning words giving certain
preimages. The first problem is whether, for a given subset, there exists a
word \emph{extending} the subset (giving a larger preimage). The second problem
is whether there exists a \emph{totally extending} word (giving the whole set
of states as a preimage)---equivalently, whether there exists an
\emph{avoiding} word for the complementary subset. The third problem is whether
there exists a \emph{resizing} word. We also consider variants where the length
of the word is upper bounded, where the size of the given subset is restricted,
and where the automaton is strongly connected, synchronizing, or binary. We
conclude with a summary of the complexities in all combinations of the cases.
