For any finite abelian group $G$ with $|G|=m$, $A\subseteq G$ and $g\in G$, let $R_{A}(g)$ be the number of solutions of the equation $g=a+b$, $a,b\in A$. Recently, Sándor and Yang [‘A lower bound of Ruzsa’s number related to the Erdős–Turán conjecture’, Preprint, 2016, arXiv:1612.08722v1] proved that, if $m\geq 36$ and $R_{A}(n)\geq 1$ for all $n\in \mathbb{Z}_{m}$, then there exists $n\in \mathbb{Z}_{m}$ such that $R_{A}(n)\geq 6$. In this paper, for any finite abelian group $G$ with $|G|=m$ and $A\subseteq G$, we prove that (a) if the number of $g\in G$ with $R_{A}(g)=0$ does not exceed $\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$, then there exists $g\in G$ such that $R_{A}(g)\geq 6$; (b) if $1\leq R_{A}(g)\leq 6$ for all $g\in G$, then the number of $g\in G$ with $R_{A}(g)=6$ is more than $\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$.