  A realization of representations of the Lie algebra $\mathfrak{o}_5$ in the
space of functions on a group $Spin_5\simeq Sp_4$ is considered. In a
representation we take a Gelfand-Tsetlin type base associated with a
restriction $\mathfrak{o}_5\downarrow\mathfrak{o}_3$. Such a base is useful is
problems appearing in quantum mechanics. We construct explicitely functions on
the group that correspond to base vectors. As in the cases of Lie algebras
$\mathfrak{gl}_3$, $\mathfrak{sp}_4$ these functions can be expressed through
$A$-hypergeometric functions (this does not hold for algebras of these series
in higher dimentions). Using this realization formulas for the action of
generators are obtained.
