In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference sets in $A_p(m,θ)$ with $p>2$. When the order of $θ$ is odd, we construct central difference sets in $A_2(m,θ)$. Furthermore, we give some reduced linking systems of difference sets in $A_2(m,θ)$ by using the difference sets we constructed. In the case $p>2$, we construct Latin square type central partial difference sets in $A_p(m,θ)$ by a similar method.