On finite CG-groups

A finite group $G$ is a CG-group if $\mathrm{\mid }\mathrm{C}\mathrm{e}\mathrm{n}\mathrm{t}(G)\mathrm{\mid }=\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }+2$, where $G^{\mathrm{\prime }}$ is the commutator subgroup and Cent$(G)$ is the set of distinct element centralizers of $G$. In this paper we give some results on CG-groups. We also give a negative answer to \cite[Conjecture 2.3]{con} given by K. Khoramshahi and M. Zarrin.