to the real galaxy sample. The mean of the residual is 0.68 and the deviation is 3.97 in units of galaxies per group or cluster, roughly $10\%$ of the maximum number count $\sim$ 45 in the real galaxy sample. The above two tests demonstrate that the background subtraction method can successfully recover the underlying group galaxy properties.

Additionally, we also estimate the impact of the projection and photo-$z$s effects on the quiescent fraction. We first simulate group galaxies with a linearly decreasing quiescent fraction with increasing radius in 3D space. We then compute the quiescent fraction as a function of the projected radius to understand the projection effect. We also perturb each galaxy redshift to simulate the photo-$z$ effect and estimate the quiescent fraction as a function of the projected radius to mimic the case we will explore in this study. In Figure \[f3\], we show a test result as an example, with the simulated quiescent fraction from 0.7 at the center to 0.3 at the boundary, 1.5 Mpc. Purple, blue, and yellow colors denote the cases using the 3D radius ($r_{3\textrm{D}}$), the projected radius, and the projected radius with a photo-$z$ inaccuracy $\sigma_{\Delta z/(1+z)} = 0.05$, respectively. Note that 