  We investigate the group structure of center-preserving automorphisms of the
finite Heisenberg group over $\mathbb Z_N$ with $U(1)$ extension, which arises
in finite-dimensional quantum mechanics on a discrete phase space. Constructing
an explicit splitting, it is shown that, for $N=2(2k+1)$, the group is
isomorphic to the semidirect product of $Sp_N$ and $\mathbb Z_N^2$. Moreover,
when N is divisible by $2^l (l \ge 2)$, the group has a non-trivial 2-cocycle,
and its explicit form is provided. By utilizing the splitting, it is
demonstrated that the corresponding projective Weil representation can be
lifted to linear representation.
