nLab
deck transformation

Definition

A deck transformation or cover automorphism is an automorphism of a cover.

i.e. if $p:E\to X$ is a cover then a cover automorphism $f\in \mathrm{deck}(p)=\{f\mid f\in \mathrm{Aut}(E),p\circ f=p\}\subseteq \mathrm{Aut}(E)$ is an automorphism of $E$ such that $p$ is invariant under composition with $f$.