Take any element $H$ and it would be acting as a $p^2$ cycle. If I remember correctly, from the natural representation of $S_n$ we can obtain two $1$-D invariant subspaces if $p=2$ (with eigenvalue $1$ and $-1$) and otherwise only one with eigenvalue $1$. Nevertheless, those would be in the $n$-dimensional natural rep and they could well correspond to $0$ in $\mathbb{F}^3_p$.
On the other hand the minimal polynomial must divide $(X^{p^2}-I)$ and cannot have degree higher than 3. If $p = 2$, then the Frobenius map gives $(x^4-1)=(x-1)^4$ so every such matrix corresponds to some upper triangular Jordan forms with eigenvalue $1$ and different block sizes, but I still do not see how we can get $1$ element to uppertriangularize all of them simultaneously. If $p>2$, there is a deg $2$ factor in $(X^{p^2}+I)$ corresponding to $\Phi_3(x)$ so we have quite a few conjugacy classes. I do not know how to proceed next.