  We study average $2$-torsion in the class group of monogenised fields of odd
degree. Bhargava-Hanke-Shankar have recently shown that the average number of
non-trivial $2$-torsion elements in the class group of monogenised cubic fields
of a fixed signature is twice the value predicted by the
Cohen-Lenstra-Martinet-Malle heuristic for the full family. Fix any odd degree
$n$ and signature $(r_{1},r_{2})$. In this work, we prove that the average
number of non-trivial elements in the class group of monogenised fields of
degree $n$ and signature $(r_{1},r_{2})$ is bounded by twice the value
predicted by Cohen-Lenstra-Martinet-Malle. Conditional on a tail estimate for
$n \ge 5$, this shows that the doubling phenomenon uncovered for cubic fields
by Bhargava-Hanke-Shankar persists across all odd degrees and signatures.
