  Understanding the structure of a graph along with the structure of its
subgraphs is important for several problems in graph theory. Two examples are
the Reconstruction Conjecture and isomorph-free generation. This paper raises
the question of which pairs of groups can be represented as the automorphism
groups of a graph and a vertex-deleted subgraph. This, and more surprisingly
the analogous question for edge-deleted subgraphs, are answered in the most
positive sense using concrete constructions.
