)]]{}\] Let $G$ be a finite non-abelian $p$-group. Let $M$ and $N$ be normal subgroups of $G$ such that $M\le Z(G)\le N$. Then $\operatorname{Aut}_{N}^{M}(G)=\operatorname{Aut}_c(G)$ if and only if one of the following conditions hold:  0.  $M=Z(G)$ and either $