  By Thurston's hyperbolization theorem, irreducible handlebody-knots are
classified into three classes: hyperbolic, toroidal, and atoroidal cylindrical.
It is known that a non-trivial handlebody-knot of genus two has a finite
symmetry group if and only if it is atoroidal. The paper investigates the
topology of cylindrical handlebody-knots of genus two that admit an unknotting
annulus; we show that the symmetry group is trivial if the unknotting annulus
is unique and of type $2$.
