Where did you come up with interpolating the "error"? (And how do you measure the error?)
On the first visit to a finer grid, the entire solution $u$ must be interpolated, ideally using a higher order operator (e.g., postprocessed/reconstructed solution for FEM). This FMG interpolation is $u^h \gets \mathbb{I}_H^h u^H$. (It's okay to use the a normal interpolation $\mathbb{I}_h^H = I_h^H$, but this typically gives up some efficiency, at least for smooth problems.)
After FMG interpolation, you just apply one or more V-cycles (or W-cycles, etc). (Make sure to run at least one smoother before restricting.) The most common choices are linear defect correction in which only the residual $r^h = A^h u^h - b^h$ is restricted and the Full Approximation Scheme (FAS) which is a natural for nonlinear problems because it avoids global linearization (e.g, Newton or Picard).
In FAS, the fine grid state is restricted using the state restriction operator $\tilde u^H \gets \hat I_h^H \tilde u^h$. State restriction is not required by linear defect correction multigrid (a convenient attribute). The most common state restrictions are nodal injection (for FD and FE) and coarse cell averages (for FV and mixed FE). Now we can write the FAS coarse grid equation (equally valid for nonlinear $A$) as
$$A^H u^H = \underbrace{I_h^H b^h}_{b^H} + \underbrace{A^H \hat I_h^H \tilde u^h - I_h^H A^h \tilde u^h}_{\tau_h^H}$$
where we have identified the coarse representation of the right hand side, $b^H$, and the additional correction $\tau_h^H$ which represents the influence of the fine grid on the coarse grid equation. Note the property that the restriction of the fine grid solution $u^{h*}$ satisfies the coarse grid equation: $A^H \hat I_h^H u^{h*} = b^H + \tau_h^{H*}$. After solving the coarse grid equation, FAS interpolates the change, leading to an updated fine solution $u^h \gets \tilde u^h + I_H^h (u^H - \hat I_h^H \tilde u^h)$.
