  We present the X-ray properties of 108 Dust-Obscured Galaxies (DOGs; F$_{24
\mu m}$/F$_{R} >$ 1000) in the COSMOS field, all of which detected in at least
three far-infrared bands with the Herschel Observatory. Out of the entire
sample, 22 are individually detected in the hard 2-8 keV X-ray band by the
Chandra COSMOS Legacy survey, allowing us to classify them as AGN. Of them, 6
(27%) are Compton Thick AGN candidates with column densities
N$_{H}$$>$10$^{24}$ cm$^{-2}$ while 15 are moderately obscured AGNs with
10$^{22}$ $<$ N$_{H}$ $<$ 10$^{24}$ cm$^{-2}$. Additionally, we estimate AGN
contributions to the IR luminosity (8-1000$\mu$m rest-frame) greater than 20%
for 19 DOGs based on SED decomposition using Spitzer/MIPS 24$\mu$m and the five
Herschel bands (100-500 $\mu$m). Only 7 of these are detected in X-rays
individually. We performed a X-ray stacking analysis for the 86 undetected
DOGs. We find that the AGN fraction in DOGs increases with 24$\mu$m flux and
that it is higher than that of the general 24$\mu$m population. However, no
significant difference is found when considering only X-ray detections. This
strongly motivates the combined use of X-ray and far-IR surveys to successfully
probe a wider population of AGNs, particularly for the most obscured ones.
