The purpose of this paper is to classify invariant hypercomplex structures on a
4
4
-dimensional real Lie group
G
G
. It is shown that the
4
4
-dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group
H
\mathbb H
of the quaternions, the multiplicative group
H
∗
{\mathbb H}^*
of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces,
R
H
4
{\mathbb R}H^4
and
C
H
2
{\mathbb C}H^2
, respectively, and the semidirect product
C
⋊
C
{\mathbb C}\rtimes {\mathbb C}
. We show that the spaces
C
H
2
{\mathbb C}H^2
and
C
⋊
C
{\mathbb C}\rtimes {\mathbb C}\,
possess an
R
P
2
{\mathbb R}P^2
of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian
4
4
-manifolds are determined.