  Let $X$ be a smooth projective hypersurface of dimension at least $3$ and
degree $2 \leq d \leq 2N - 3$. Assume that either $X$ is Fano with $d \leq 8$,
or $X$ is general. We show that every automorphism of the Hilbert square
$X^{[2]}$ of $X$ is induced by some automorphism of $X$.
