  Let $X$ be a compact quotient of the product of the real Heisenberg group
$H_{4m+1}$ of dimension $4m+1$ and the 3-dimensional real Euclidean space
$\bR^3$. A left invariant hypercomplex structure on $H_{4m+1}\times \bR^3$
descends onto the compact quotient $X$. The space $X$ is a hyperholomorphic
fibration of 4-tori over a $4m$-torus. We calculate the parameter space and
obstructions to deformations of this hypercomplex structure on $X$. Using our
calculations we show that all small deformations generate invariant
hypercomplex structures on $X$ but not all of them arise from deformations of
the lattice. This is in contrast to the deformations on the $4m$-torus.
