  We consider and resolve the gap problem for almost quaternion-Hermitian
structures, i.e. we determine the maximal and submaximal symmetry dimensions,
both for Lie algebras and Lie groups, in the class of almost
quaternion-Hermitian manifolds. We classify all structures with such symmetry
dimensions. Geometric properties of the submaximally symmetric spaces are
studied, in particular we identify locally conformally quaternion-K\"ahler
structures as well as quaternion-K\"ahler with torsion.
