  Compared with numerous X-ray dominant active galactic nuclei (AGNs) without
emission-line signatures in their optical spectra, the X-ray selected AGNs with
optical emission lines are probably still in the high-accretion phase of black
hole growth. This paper presents an investigation on the fraction of these
X-ray detected AGNs with optical emission-line spectra in 198 galaxy groups at
$z<1$ in a rest frame 0.1-2.4 keV luminosity range 41.3 <log(L_X/erg s-1) <
44.1 within the COSMOS field, as well as its variations with redshift and group
richness. For various selection criteria of member galaxies, the numbers of
galaxies and the AGNs with optical emission lines in each galaxy group are
obtained. It is found that, in total 198 X-ray groups, there are 27 AGNs
detected in 26 groups. AGN fraction is on everage less than $4.6 (\pm 1.2)\%$
for individual groups hosting at least one AGN. The corrected overall AGN
fraction for whole group sample is less than $0.98 (\pm 0.11) \%$. The
normalized locations of group AGNs show that 15 AGNs are found to be located in
group centers, including all 6 low-luminosity group AGNs. A week rising
tendency with $z$ are found: overall AGN fraction is 0.30-0.43% for the groups
at $z<0.5$, and 0.55-0.64% at 0.5 < z < 1.0. For the X-ray groups at $z>0.5$,
most member AGNs are X-ray bright, optically dull, which results in a lower AGN
fractions at higher redshifts. The AGN fraction in isolated fields also
exhibits a rising trend with redshift, and the slope is consistent with that in
groups. The environment of galaxy groups seems to make no difference in
detection probability of the AGNs with emission lines. Additionally, a larger
AGN fractions are found in poorer groups, which implies that the AGNs in poorer
groups might still be in the high-accretion phase, whereas the AGN population
in rich clusters is mostly in the low-accretion, X-ray dominant phase.
