  We present a detailed study of the variable star population of Eridanus II
(Eri II), an ultra-faint dwarf galaxy that lies close to the Milky Way virial
radius. We analyze multi-epoch $g,r,i$ ground-based data from Goodman and the
Dark Energy Camera, plus $F475W, F606W, F814W$ space data from the Advanced
Camera for Surveys. We report the detection of 67 RR Lyrae (RRL) stars and 2
Anomalous Cepheids, most of them new discoveries. With the RRL stars, we
measure the distance modulus of Eri II, $\mu_0=22.84\pm 0.05$ mag
(D$_{\odot}=370\pm9$ kpc) and derive a metallicity spread of 0.3 dex (0.2 dex
intrinsic). The colour distribution of the horizontal branch (HB) and the
period distribution of the RRL stars can be nicely reproduced by a combination
of two stellar models of [Fe/H]=($-2.62$, $-2.14$). The overall low metallicity
is consistent with the red giant branch bump location, 0.65 mag brighter than
the HB. These results are in agreement with previous spectroscopic studies. The
more metal-rich RRL and the RRab stars have greater central concentration than
the more metal-poor RRL and the RRc stars that are mainly located outside $\sim
1$ r$_{\rm h}$. This is similar to what is found in larger dwarf galaxies such
as Sculptor, and in agreement with an outside-in galaxy formation scenario.
This is remarkable in such a faint dwarf galaxy with an apparently single and
extremely short ($<1$ Gyr) star formation burst. Finally, we have derived new
and independent structural parameters for Eri II and its star cluster using our
new data that are in very good agreement with previous estimates.
