In this article we investigate the question which local symmetry preserving operations can not only preserve, but also increase the symmetry of a polyhedral map. Often operations that can increase symmetry, can nevertheless not do so for polyhedral maps of every genus. So for maps that can increase symmetry, we also investigate for which genera they can do so. We give complete answers for operations with inflation factor at most 6 (that is: that increase the number of edges by a factor of at most 6) and for the chemically relevant Goldberg-Coxeter operations and the leapfrog operation.