  In this paper, we explore Fourier analysis for noncommutative $L_p$
space-valued functions on $G$, where $G$ is a totally disconnected non-abelian
compact group. By additionally assuming that the value of these functions
remains invariant within each conjugacy class, we establish a noncommutative
maximal inequality for Fej\'er means utilizing the associated character system
of $G$. This is an operator-valued version of the classical result due to
G\'at. We follow essentially the classical sketch, but due to the
noncommutativity, many classical arguments have to be revised. Notably,
compared to the classical results. the bounds of our estimates are explicity
calculated.
