Using decays to $phi$-meson pairs, the inclusive production of charmonium states in $b$-hadron decays is studied with $pp$ collision data corresponding to an integrated luminosity of $3.0fb^{-1}$, collected by the LHCb experiment at centre-of-mass energies of 7 and 8 TeV. Denoting by $B_Cequiv B(bto CX)times B(Ctophiphi)$ the inclusive branching fraction of a $b$ hadron to a charmonium state $C$ that decays into a pair of $phi$ mesons, ratios $R^{C1}_{C2}equiv B_{C1}/B_{C2}$ are determined as $R^{chi_{c0}}_{eta_c(1S)}=0.147pm0.023pm0.011$, $R^{chi_{c1}}_{eta_c (1S)}=0.073pm0.016pm0.006$, $R^{chi_{c2}}_{eta_c (1S)}=0.081pm0.013pm0.005$, $R^{chi_{c1}}_{chi_{c0}}=0.50pm0.11pm0.01$, $R^{chi_{c2}}_{chi_{c0}}=0.56pm0.10pm0.01$ and $R^{eta_c (2S)}_{eta_c(1S)}=0.040pm0.011pm0.004$. Here and below the first uncertainties are statistical and the second systematic. Upper limits at $90%$ confidence level for the inclusive production of $X(3872)$, $X(3915)$ and $chi_{c2}(2P)$ states are obtained as $R^{X(3872)}_{chi_{c1}}<0.34$, $R^{X(3915)}_{chi_{c0}}<0.12$ and $R^{chi_{c2}(2P)}_{chi_{c2}}<0.16$. Differential cross-sections as a function of transverse momentum are measured for the $eta_c(1S)$ and $chi_c$ states. The branching fraction of the decay $B_s^0rightarrowphiphiphi$ is measured for the first time, $B(B_s^0tophiphiphi)=(2.15pm0.54pm0.28pm0.21_{B})times 10^{-6}$. Here the third uncertainty is due to the branching fraction of the decay $B_s^0tophiphi$, which is used for normalization. No evidence for intermediate resonances is seen. A preferentially transverse $phi$ polarization is observed. The measurements allow the determination of the ratio of the branching fractions for the $eta_c(1S)$ decays to $phiphi$ and $pbar{p}$ as $B(eta_c(1S)tophiphi)/B(eta_c(1S)to pbar{p})=1.79pm0.14pm0.32$.
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