let $G$ be a finite group with $1\neq Z(G) \lneqq G$. Also let $H=\{x_1,...,x_n\}$ be the set of all disjoint representative elements of right cosets of $Z(G)$ in $G$.
Is there any non-trivial element in $Aut(G)$, call it $\phi$, with the property that for each $x_i\in H$ we have $\phi(Z(G)x_i)=Z(G)x_i$ i.e. $\phi$ maps each element belongs to $Z(G)x_i$ to an element in $Z(G)x_i$? What can we say if $G$ be a finite $p$-group?
Thanks in advance for any helpful guide.