  A detailed study of the $g$-factor anisotropy of electrons and holes in
InAs/In$_{0.53}$Al$_{0.24}$Ga$_{0.23}$As self-assembled quantum dots emitting
in the telecom spectral range of $1.5-1.6$ $\mu$m (around 0.8 eV photon energy)
is performed by time-resolved pump-probe ellipticity technique using a
superconducting vector magnet. All components of the $g$-factor tensors are
measured, including their spread in the quantum dot (QD) ensemble.
Surprisingly, the electron $g$ factor shows a large anisotropy changing from
$g_{\mathrm{e},x}= -1.63$ to $g_{\mathrm{e},z}= -2.52$ between directions
perpendicular and parallel to the dot growth axis, respectively, at an energy
of 0.82 eV. The hole $g$-factor anisotropy at this energy is even stronger:
$|g_{\text{h},x}|= 0.64$ and $|g_{\text{h},z}|= 2.29$. On the other hand, the
in-plane anisotropies of electron and hole $g$ factors are small. The
pronounced out-of-plane anisotropy is also observed for the spread of the $g$
factors, determined from the spin dephasing time. The hole longitudinal $g$
factors are described with a theoretical model that allows us to estimate the
QD parameters. We find that the QD height-to-diameter ratio increases while the
indium composition decreases with increasing QD emission energy.
