This is prompted by my attempts to work on this question. Let $H \subset G \subseteq S_d$ be transitive permutation groups. Recall that an element of $S_d$ is called a derangement if it has no fixed points.
Is the proportion of derangements in $H$ always greater than in $G$?
If $H$ doesn't have to be transitive, then the answer is "no"; just let $H$ be trivial. But a quick sampling of examples with $G$ and $H$ both transitive doesn't turn up any counterexamples.
UPDATE Never mind. $A_4$ inside $S_4$, the probability of a derangement goes down from $3/8$ to $1/4$.