On symmetry and exterior problems of knotted handlebodies

The paper concerns two classical problems in knot theory pertaining to knot symmetry and knot exteriors. In the context of a knotted handlebody $V$ in a $3$-sphere $S^3$, the symmetry problem seeks to classify the mapping class group of the pair $(S^3,V)$, whereas the exterior problem examines to what extent the exterior $E(V)$ determines or fails to determine the isotopy type of $V$. The paper determines the symmetries of knotted genus two handlebodies arising from hyperbolic knots with non-integral toroidal Dehn surgeries, and solve the knot exterior problem for them. A new interpretation and generalization of a Lee-Lee family of knotted handlebodies is provided.