A note on the codegree of finite groups

Let $\chi$ be an irreducible character of a group $G,$ and $S_c(G)={\sum }_{\chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G)}\mathrm{c}\mathrm{o}\mathrm{d}(\chi )$ be the sum of the codegrees of the irreducible characters of $G\mathrm{.}$ Write $\mathrm{f}\mathrm{c}\mathrm{o}\mathrm{d}(G)=\frac{S_c(G)}{\mathrm{\mid }G\mathrm{\mid }}\mathrm{.}$ We aim to explore the structure of finite groups in terms of $\mathrm{f}\mathrm{c}\mathrm{o}\mathrm{d}(G)\mathrm{.}$ On the other hand, we determine the lower bound of $S_c(G)$ for nonsolvable groups and prove that if $G$ is nonsolvable, then $S_c(G)\ge S_c(A_5)=68,$ with equality if and only if $G\cong A_5\mathrm{.}$ Additionally, we show that there is a solvable group so that it has the codegree sum as $A_5\mathrm{.}$