  Sterile neutrino is a promising dark matter (DM) candidate. However the
parameter space of this scenario has almost been ruled out by the X-ray
observation results whenever the sterile neutrino is solely produced by the
Dodelson-Widrow (DW) mechanism in the early Universe. In this letter we propose
an extension to the minimal sterile neutrino DM model by introducing the
pseudo-Dirac sterile neutrino, which implies the existence of two nearly
degenerate Majorana states $\hat N_{1,2}^{}$, and a permutation symmetry. The
heavy state $\hat N_1$ is produced via the DW mechanism, and the light state
$\hat N_2$, which serves as the DM, is produced from the decay of $\hat N_1$ in
the early Universe. The X-ray constraint is avoided by the permutation
symmetry, which forbidden the two-body decay of the DM into active neutrinos
and photon. A promising signal of this scenario is the effective number of
neutrino species, which will be precisely measured in future experiments, such
as CMB stage IV. We further study the impact of this model on the cosmological
parameters. The Markov Chain Monte Carlo analysis for the Planck + BAO+R19 data
gives $H_0=69.2_{-0.59}^{+0.58}$, which may relieve the Hubble tension.
