  We present results on transverse momentum ($p_{\rm T}$) and rapidity ($y$)
differential production cross sections, mean transverse momentum and mean
transverse momentum square of inclusive J/$\psi$ and $\psi(2S)$ at forward
rapidity ($2.5<y<4$) as well as $\psi(2S)$-to-J/$\psi$ cross section ratios.
These quantities are measured in pp collisions at center of mass energies
$\sqrt{s}=5.02$ and 13 TeV with the ALICE detector. Both charmonium states are
reconstructed in the dimuon decay channel, using the muon spectrometer. A
comprehensive comparison to inclusive charmonium cross sections measured at
$\sqrt{s}=2.76$, 7 and 8 TeV is performed. A comparison to non-relativistic
quantum chromodynamics and fixed-order next-to-leading logarithm calculations,
which describe prompt and non-prompt charmonium production respectively, is
also presented. A good description of the data is obtained over the full
$p_{\rm T}$ range, provided that both contributions are summed. In particular,
it is found that for $p_{\rm T}>15$ GeV/$c$ the non-prompt contribution reaches
up to 50% of the total charmonium yield.
