  Let $\mathscr Q$ be the quaternion Heisenberg group, and let $\mathbf P$ be
the affine automorphism group of $\mathscr Q$. We develop the theory of
continuous wavelet transform on the quaternion Heisenberg group via the unitary
representations of $\mathbf P$ on $L^2(\mathscr Q)$. A class of radial wavelets
is constructed. The inverse wavelet transform is simplified by using radial
wavelets. Then we investigate the Radon transform on $\mathscr Q$. A
Semyanistri-Lizorkin space is introduced, on which the Radon transform is a
bijection. We deal with the Radon transform on $\mathscr Q$ both by the
Euclidean Fourier transform and the group Fourier transform. These two
treatments are essentially equivalent. We also give an inversion formula by
using wavelets, which does not require the smoothness of functions if the
wavelet is smooth.
