Automorphism group of the graph $A(n,k,r)$

Let [n]⁽ᵏ⁾ be the set of all ordered k-tuples of distinct elements in [n]=\1,2,...,n\. The $(n,k,r)$-arrangement graph $A(n,k,r)$ with 1≤ r≤ k≤ n, is the graph with vertex set [n]⁽ᵏ⁾ and with two k-tuples are adjacent if they differ in exactly r coordinates. In this manuscript, we characterize the full automorphism groups of $A(n,k,r)$ in the cases that 1≤ r=k≤ n and $r=2<k=n$. Thus, we resolve two special cases of an open problem proposed by Fu-Gang Yin, Yan-Quan Feng, Jin-Xin Zhou and Yu-Hong Guo. In addition, we conclude with a bold conjecture.