Large sum-free sets in Z5n

It is well-known that for a prime $p\equiv 2\left(\mathrm{mod}3\right)$ and integer $n\ge 1$, the maximum possible size of a sum-free subset of the elementary abelian group $Z_p^n$ is $\frac{1}{3}(p+1)p^{n-1}$. However, the matching stability result is known for $p=2$ only. We consider the first open case $p=5$ showing that if $A\subseteq Z_5^n$ is a sum-free subset with $\left|A\right|>\frac{3}{2}\cdot 5^{n-1}$, then there are a subgroup $H<Z_5^n$ of size $\left|H\right|=5^{n-1}$ and an element $e\notin H$ such that $A\subseteq (e+H)\cup (-e+H)$.