  The transverse momentum ($p_{\rm T}$) dependence of the nuclear modification
factor $R_{\rm AA}$ and the centrality dependence of the average transverse
momentum $\langle p_{\rm T}\rangle$ for inclusive J/$\psi$ have been measured
with ALICE for Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV in the
e$^+$e$^-$ decay channel at mid-rapidity ($|y|<0.8$). The $\langle p_{\rm
T}\rangle$ is significantly smaller than the one observed for pp collisions at
the same centre-of-mass energy. Consistently, an increase of $R_{\rm AA}$ is
observed towards low $p_{\rm T}$. These observations might be indicative of a
sizable contribution of charm quark coalescence to the J/$\psi$ production.
Additionally, the fraction of non-prompt J/$\psi$ from beauty hadron decays,
$f_{\rm B}$, has been determined in the region $1.5 < p_{\rm T} < 10$ GeV/c in
three centrality intervals. No significant centrality dependence of $f_{\rm B}$
is observed. Finally, the $R_{\rm AA}$ of non-prompt J/$\psi$ is discussed and
compared with model predictions. The nuclear modification in the region $4.5 <
p_{\rm T} < 10$ GeV/c is found to be stronger than predicted by most models.
