Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$.