  We report an infrared spectroscopy study of the axion topological insulator
candidate EuIn$_2$As$_2$ for which the Eu moments exhibit an A-type
antiferromagnetic (AFM) order below $T_N \simeq 18 \mathrm{K}$. The low energy
response is composed of a weak Drude peak at the origin, a pronounced
infrared-active phonon mode at 185 cm$^{-1}$ and a free carrier plasma edge
around 600 cm$^{-1}$. The interband transitions start above 800 cm$^{-1}$ and
give rise to a series of weak absorption bands at 5\,000 and 12\,000 cm$^{-1}$
and strong ones at 20\,000, 27\,500 and 32\,000 cm$^{-1}$. The AFM transition
gives rise to pronounced anomalies of the charge response in terms of a
cusp-like maximum of the free carrier scattering rate around $T_N$ and large
magnetic splittings of the interband transitions at 5\,000 and 12\,000
cm$^{-1}$. The phonon mode at 185 cm$^{-1}$ has also an anomalous temperature
dependence around $T_N$ which suggests that it couples to the fluctuations of
the Eu spins. The combined data provide evidence for a strong interaction
amongst the charge, spin and lattice degrees of freedom.
