  Inclusive J/$\psi$ production is studied in p-Pb interactions at a
centre-of-mass energy per nucleon-nucleon collision $\sqrt{s_{\rm NN}}=8.16$
TeV, using the ALICE detector at the CERN LHC. The J/$\psi$ meson is
reconstructed, via its decay to a muon pair, in the centre-of-mass rapidity
intervals $2.03<y_{\rm {cms}}<3.53$ and $-4.46<y_{\rm {cms}}<-2.96$, where
positive and negative $y_{\rm {cms}}$ refer to the p-going and Pb-going
direction, respectively. The transverse momentum coverage is $p_{\rm T}<20$
GeV/$c$. In this paper, $y_{\rm cms}$- and $p_{\rm T}$-differential cross
sections for inclusive J/$\psi$ production are presented, and the corresponding
nuclear modification factors $R_{\rm pPb}$ are shown. Forward results show a
suppression of the J/$\psi$ yield with respect to pp collisions, concentrated
in the region $p_{\rm T}\lesssim 5$ GeV/$c$. At backward rapidity no
significant suppression is observed. The results are compared to previous
measurements by ALICE in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV and to
theoretical calculations. Finally, the ratios $R_{\rm FB}$ between forward- and
backward-$y_{\rm {cms}}$ $R_{\rm pPb}$ values are shown and discussed.
