Let $A$ be an abelian group generated by a $2$-element set $S=\{a, b: a^m=b^n=e, m,n\ge 2\}$, where $e$ is the identity element of $A$. Let $\Gamma_{m,n}=Cay_g(A, S)$ be the undirected Cayley graph of $A$ associated with $S$. In this paper, it is shown that $\Gamma_{2k+1,2l+1}$, $\Gamma_{2, 2+l}$ and $\Gamma_{2k+1, 6}$ are Semi Strong OUTER Mod Sum Graphs, and $\Gamma_{k, l}$ is Anti-OUTER Mod Sum Graph, for every $k,l\in \mathbb{Z}^+$.