Dividing a point by a specific number in elliptic curve
Find an integer $k$ such that $65537k \equiv 1 \pmod q$; such a $k$ exists because $q$ is prime.

If $65537\cdot P = H$, then clearly $P = k\cdot H$ (keeping in mind that $q\cdot P = \mathcal{O}$ for all points $P$), and conversely $(k\cdot H)\cdot 65537 = H$.