  We construct manifestly ${\cal N}=4$ supersymmetric off-shell superfield
actions for the HKT $d=1$ sigma models on the group manifolds U(2) and SU(3),
using the harmonic $d=1$ approach. The underlying $({\bf 4, 4, 0})$ and $({\bf
4, 4, 0})\oplus({\bf 4, 4, 0})$ multiplets are described, respectively, by one
and two harmonic analytic superfields $q^+$ satisfying the appropriate
nonlinear harmonic constraints. The invariant actions in both cases are
bilinear in the superfields. We present the corresponding superfield
realizations of the U(2) and SU(3) isometries and show that in fact they are
enlarged to the products U(2)$\times$SU(2) and SU(3)$\times$U(2). We prove the
corresponding invariances at both the superfield and component levels and
present the bosonic $d=1$ sigma model actions, as integral over $t$ in the U(2)
case and over $t$ and ${\rm SU}(2)$ harmonics in the SU(3) case. In the U(2)
case we also give a detailed comparison with the general harmonic approach to
HKT models and establish a correspondence with a particular action of the
off-shell nonlinear multiplet $({\bf 3, 4, 1})$. A possible way of generalizing
U(2) model to the matrix U($2n$) case is suggested.
