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.
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