I am trying to learn results related to the presentation complex of a group and I am new to this subject. So I apologize if the question is silly. Given a finite group $G$, and a presentation $P$ of $G$, consider the presentation complex $X_P$. I have computed using Mayer-Vietoris sequence for the group $S_3$ (the symmetric group of order $6$) and get $H_2(X_P,\mathbb{Z})\neq 0$. Also, by using a similar method I was able to show that for finite cyclic groups $H_2(X_P,\mathbb{Z})=0$. I have the following question:

Can we put some restriction on $G$ (here $G$ is a finite group) so that $H_2(X_P,\mathbb{Z})\neq 0$?