  A Theorem of Kirichenko states that the torsion 3-form of the characteristic
connection of a nearly K\"ahler manifold is parallel. On the other side, any
almost hermitian manifold of type $\mathrm{G}_1$ admits a unique connection
with totally skew symmetric torsion. In dimension six, we generalize
Kirichenko's Theorem and we describe almost hermitian $\mathrm{G}_1$-manifolds
with parallel torsion form. In particular, among them there are only two types
of $\mathcal{W}_3$-manifolds with a non-abelian holonomy group, namely twistor
spaces of 4-dimensional self-dual Einstein manifolds and the invariant
hermitian structure on the Lie group $\mathrm{SL}(2, \C)$. Moreover, we
classify all naturally reductive hermitian $\mathcal{W}_3$-manifolds with small
isotropy group of the characteristic torsion.
