  We construct an extended version of the linear sigma model in such a way as
to describe spin-$1$ hadrons as well as spin-$0$ hadrons in two-color QCD
(QC$_2$D) by respecting the Pauli-G\"{u}rsey $SU(4)$ symmetry. Within a
mean-field approximation, we therefrom examine a mass spectrum of the spin-$1$
hadrons at finite quark chemical potential ($\mu_q$) and zero temperature. Not
only mean fields of scalar mesons and scalar-diquark baryons but also of vector
mesons and vector-diquark baryons are incorporated. As a result, we find that,
unless all of those four types of mean fields are taken into account, neither
lattice result for the critical $\mu_q$ that corresponds to the onset of baryon
superfluidity nor for $\mu_q$ dependence of the pion mass can be reproduced. We
also find that a slight suppression of the $\rho$ meson mass in the superfluid
phase, which was suggested by the lattice simulation, is reproduced by subtle
mixing effects between spin-$0$ and spin-$1$ hadrons. Moreover, we demonstrate
the emergence of an axialvector condensed phase and possibly of a vector
condensed phase by identifying the values of $\mu_q$ at which the corresponding
hadron masses vanish. The possible presence of iso-triplet $1^-$ diquarks that
may be denoted by a tensor-type quark bilinear field is also discussed.
