Let $G$ be a group acting on a set $A$. Let $N$ be a non-trivial normal subgroup of $G$. Suppose that $S$ is an $N$-invariant set, i.e. $n \cdot s \in S$ for all $s \in S$, $n \in N$. Must $S$ be $G$-invariant?
I suspect the answer is yes (for example, it is true in the situation of a Galois group acting on a Galois field extension). I've been playing around with it for a while and haven't been able to get anything.